The Database of Integer Sequences, Part 6 

     Part of the On-Line Encyclopedia of Integer Sequences


This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

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    home page:          www.research.att.com/~njas/
    


(start)



%I A110858
%S A110858 1,1,1,1,2,1,1,1,2,7,1,1,1,3,13,3,938,41
%N A110858 Number of strictly Type I binary self-dual codes of length 2n.
%D A110858 W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
%H A110858 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%Y A110858 Adjacent sequences: A110855 A110856 A110857 this_sequence A110859 A110860 A110861
%Y A110858 Sequence in context: A061545 A126886 A105685 this_sequence A125090 A073266 A125692
%K A110858 nonn
%O A110858 1,5
%A A110858 njas, Sep 19 2005

%I A125090
%S A125090 1,1,0,1,1,1,1,2,1,1,1,3,0,3,0,1,4,2,5,2,1,1,5,5,6,7,2,1,1,6,9,5,15,0,5,0,
%T A125090 1,7,14,1,25,9,12,3,1,1,8,20,7,35,29,18,15,3,1,1,9,27,20,42,63,14,42,0,7,
%U A125090 0,1,10,35,39,42,112,14,85,24,22,4,1,1,11,44,65,30,174,84,134,95,40,26,4
%V A125090 1,1,0,1,-1,-1,1,-2,-1,1,1,-3,0,3,0,1,-4,2,5,-2,-1,1,-5,5,6,-7,-2,1,1,-6,9,5,-15,0,5,0,
%W A125090 1,-7,14,1,-25,9,12,-3,-1,1,-8,20,-7,-35,29,18,-15,-3,1,1,-9,27,-20,-42,63,14,-42,0,7,
%X A125090 0,1,-10,35,-39,-42,112,-14,-85,24,22,-4,-1,1,-11,44,-65,-30,174,-84,-134,95,40,-26,-4
%N A125090 Triangle read by rows: T(0,0)=1; for 0<=k<=n, n>=1, T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the tridiagonal n X n matrix with diagonal (0,1,1,...) and super- and subdiagonals (1,1,1,...).
%C A125090 The characteristic polynomial of the n X n matrix has a root = 1+2*cos(2*Pi/(2n+1)).
%F A125090 f(n,x)=(x-1)f(n-1,x)-f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1.
%e A125090 Triangle starts:
%e A125090 1;
%e A125090 1, 0;
%e A125090 1, -1, -1;
%e A125090 1, -2, -1, 1;
%e A125090 1, -3, 0, 3, 0;
%e A125090 1, -4, 2, 5, -2, -1;
%e A125090 1, -5, 5, 6, -7, -2, 1;
%e A125090 1, -6, 9, 5, -15, 0, 5, 0;
%p A125090 with(linalg): m:=proc(i,j): if i=1 and j=1 then 0 elif i=j then 1 elif abs(i-j)=1 then 1 else 0 fi end: T:=proc(n,k) if n=0 and k=0 then 1 else coeff(charpoly(matrix(n,n,m),x),x,n-k) fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A125090 Cf. A104562.
%Y A125090 Adjacent sequences: A125087 A125088 A125089 this_sequence A125091 A125092 A125093
%Y A125090 Sequence in context: A126886 A105685 A110858 this_sequence A073266 A125692 A128258
%K A125090 sign,tabl
%O A125090 1,8
%A A125090 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 19 2006
%E A125090 Edited by njas, Nov 29 2006

%I A073266
%S A073266 1,1,1,0,2,1,1,1,3,1,0,2,3,4,1,0,2,4,6,5,1,0,0,6,8,10,6,1,1,1,3,13,15,
%T A073266 15,7,1,0,2,3,12,25,26,21,8,1,0,2,6,10,31,45,42,28,9,1,0,0,6,16,30,66,
%U A073266 77,64,36,10,1,0,2,4,18,40,76,126,126,93,45,11,1,0,0,6,16,50,96,168
%N A073266 Triangle read by rows: T(n,k) is the number of compositions of n as the sum of k integral powers of 2.
%C A073266 Upper triangular region of the table A073265 read by rows. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 04 2005
%D A073266 S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
%F A073266 T(n, k)=coefficient of x^n in the formal power series (x+x^2+x^4+x^8+x^16+...)^k. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 04 2005
%F A073266 T(0, k) = T(n, 0) = 0, T(n, k) = 0 if k > n, T(n, 1) = 1 if n = 2^m, 0 otherwise, and in other cases T(n, k) = Sum_{i=0..[log2(n-1)]} T(n-(2^i), k-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 04 2005
%F A073266 Sum_{k, 0<=k<=n}T(n,k)=A023359(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 04 2006
%e A073266 T(6,3) = 4 because there are four ordered partitions of 6 into 3 powers of 2, namely: 4+1+1, 1+4+1, 1+1+4 and 2+2+2.
%e A073266 Triangle begins:
%e A073266 1;
%e A073266 1,1;
%e A073266 0,2,1;
%e A073266 1,1,3,1;
%e A073266 0,2,3,4,1;
%e A073266 0,2,4,6,5,1;
%Y A073266 Adjacent sequences: A073263 A073264 A073265 this_sequence A073267 A073268 A073269
%Y A073266 Sequence in context: A105685 A110858 A125090 this_sequence A125692 A128258 A104967
%K A073266 nonn,tabl
%O A073266 1,5
%A A073266 Antti Karttunen Jun 25 2002

%I A125692
%S A125692 1,1,1,0,2,1,1,1,3,1,0,2,3,4,1,2,2,2,6,5,1,0,4,6,0,10,6,1,5,5,3,11,5,15,
%T A125692 7,1,0,10,15,4,15,14,21,8,1,14,14,6,26,19,15,28,28,9,1,0,28,42,16,30,42,
%U A125692 7,48,36,10,1
%V A125692 1,-1,1,0,-2,1,1,1,-3,1,0,2,3,-4,1,-2,-2,2,6,-5,1,0,-4,-6,0,10,-6,1,5,5,-3,-11,-5,15,
%W A125692 -7,1,0,10,15,4,-15,-14,21,-8,1,-14,-14,6,26,19,-15,-28,28,-9,1,0,-28,-42,-16,30,42,-7,
%X A125692 -48,36,-10,1
%N A125692 Riordan array (1-x*c(-x^2),x(1-x*c(-x^2)) where c(x) is the g.f. of A000108.
%C A125692 Row sums are aerated signed Catalan numbers with g.f. c(-x^2). Inverse of A105306. First column is A105523.
%e A125692 Triangle begins
%e A125692 1,
%e A125692 -1, 1,
%e A125692 0, -2, 1,
%e A125692 1, 1, -3, 1,
%e A125692 0, 2, 3, -4, 1,
%e A125692 -2, -2, 2, 6, -5, 1,
%e A125692 0, -4, -6, 0, 10, -6, 1
%Y A125692 Adjacent sequences: A125689 A125690 A125691 this_sequence A125693 A125694 A125695
%Y A125692 Sequence in context: A110858 A125090 A073266 this_sequence A128258 A104967 A098495
%K A125692 easy,sign,tabl
%O A125692 0,5
%A A125692 Paul Barry (pbarry(AT)wit.ie), Nov 30 2006

%I A128258
%S A128258 1,0,1,1,2,1,1,1,3,1,1,0,0,4,1,0,3,1,0,5,1,1,0,0,0,0,6,1,0,0,3,1,0,0,7,
%T A128258 1,0,2,1,0,0,0,0,8,1,0,1,0,4,1,0,0,0,9,1
%V A128258 1,0,1,-1,2,1,-1,-1,3,1,-1,0,0,4,1,0,-3,-1,0,5,1,-1,0,0,0,0,6,1,0,0,-3,-1,0,0,7,1,0,-2,
%W A128258 -1,0,0,0,0,8,1,0,-1,0,-4,-1,0,0,0,9,1
%N A128258 Moebius transform of A128229.
%C A128258 Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6,...). A128259 = inverse Moebius transform of A128229.
%F A128258 A054525 * A128229
%e A128258 First few rows of the triangle are:
%e A128258 1;
%e A128258 0, 1;
%e A128258 -1, 2, 1;
%e A128258 -1, -1, 3, 1;
%e A128258 -1, 0, 0, 4, 1;
%e A128258 0, -3, -1, 0, 5, 1;
%e A128258 ...
%Y A128258 Cf. A128259, A128229, A054525, A000010.
%Y A128258 Adjacent sequences: A128255 A128256 A128257 this_sequence A128259 A128260 A128261
%Y A128258 Sequence in context: A125090 A073266 A125692 this_sequence A104967 A098495 A095025
%K A128258 tabl,sign
%O A128258 1,5
%A A128258 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 21 2007

%I A104967
%S A104967 1,1,1,1,2,1,1,1,3,1,1,0,0,4,1,1,1,2,2,5,1,1,2,3,4,5,6,1,1,3,3,3,5,9,7,
%T A104967 1,1,4,2,0,0,4,14,8,1,1,5,0,4,6,6,0,20,9,1,1,6,3,8,10,12,14,8,27,10,1,1,
%U A104967 7,7,11,10,10,14,22,21,35,11,1,1,8,12,12,5,0,0,8,27,40,44,12,1
%V A104967 1,-1,1,-1,-2,1,-1,-1,-3,1,-1,0,0,-4,1,-1,1,2,2,-5,1,-1,2,3,4,5,-6,1,-1,3,3,3,5,9,-7,1,
%W A104967 -1,4,2,0,0,4,14,-8,1,-1,5,0,-4,-6,-6,0,20,-9,1,-1,6,-3,-8,-10,-12,-14,-8,27,-10,1,-1,
%X A104967 7,-7,-11,-10,-10,-14,-22,-21,35,-11,1,-1,8,-12,-12,-5,0,0,-8,-27,-40,44,-12,1
%N A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.
%C A104967 Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.
%F A104967 G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
%e A104967 Triangle begins:
%e A104967 1;
%e A104967 -1,1;
%e A104967 -1,-2,1;
%e A104967 -1,-1,-3,1;
%e A104967 -1,0,0,-4,1;
%e A104967 -1,1,2,2,-5,1;
%e A104967 -1,2,3,4,5,-6,1;
%e A104967 -1,3,3,3,5,9,-7,1;
%e A104967 -1,4,2,0,0,4,14,-8,1;
%e A104967 -1,5,0,-4,-6,-6,0,20,-9,1; ...
%o A104967 (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)),n,x),k,y)}
%Y A104967 Cf. A090132, A104969, A104969.
%Y A104967 Adjacent sequences: A104964 A104965 A104966 this_sequence A104968 A104969 A104970
%Y A104967 Sequence in context: A073266 A125692 A128258 this_sequence A098495 A095025 A069897
%K A104967 sign,tabl
%O A104967 0,5
%A A104967 Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005

%I A098495
%S A098495 1,1,0,1,1,1,1,2,1,1,1,3,1,1,0,1,4,5,1,1,1,1,5,11,7,2,1,1,1,6,19,29,9,1,
%T A098495 1,0,1,7,29,71,76,11,1,1,1,1,8,41,139,265,199,13,2,1,1,1,9,55,239,666,
%U A098495 989,521,15,1,1,0,1,10,71,377,1393,3191,3691,1364,17,1,1,1,1,11,89,559
%V A098495 1,1,0,1,-1,-1,1,-2,-1,-1,1,-3,1,1,0,1,-4,5,1,1,1,1,-5,11,-7,-2,-1,1,1,-6,19,-29,9,1,
%W A098495 -1,0,1,-7,29,-71,76,-11,1,1,-1,1,-8,41,-139,265,-199,13,-2,1,-1,1,-9,55,-239,666,-989,
%X A098495 521,-15,1,-1,0,1,-10,71,-377,1393,-3191,3691,-1364,17,1,-1,1,1,-11,89,-559
%N A098495 Array T(r,c) read by antidiagonals: values of triangle A098493 interpreted as polynomials, r>=0.
%D A098495 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
%F A098495 Recurrence: T(r, 1) = 1, T(r, 2) = -r-1, T(r, c) = -rT(r, c-1) - T(r, c-2). (Corrected Oct 19 2004)
%e A098495 1,0,-1,-1,0,1,1,0,-1,
%e A098495 1,-1,-1,1,1,-1,-1,1,1,
%e A098495 1,-2,1,1,-2,1,1,-2,1,
%e A098495 1,-3,5,-7,9,-11,13,-15,
%e A098495 1,-4,11,-29,76,-199,521,
%e A098495 1,-5,19,-71,265,-989,3691,
%t A098495 T[r_, 1] := 1; T[r_, 2] := -r - 1; T[r_, c_] := -r*T[r, c - 1] - T[r, c - 2]; Flatten[ Table[ T[n - i, i], {n, 0, 11}, {i, n + 1}]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 10 2005)
%o A098495 (PARI) { t(r,c)=if(c>r||c<0||r<0,0,if(c>=r-1,(-1)^r*if(c==r,1,-c),if(r==1,0,if(c==0,t(r-1,0)-t(r-2,0),t(r-1,c)-t(r-2,c)-t(r-1,c-1))))) } T(r,c)=sum(i=0,c,t(c,i)*r^i)
%Y A098495 See A094954 (with negative k) for negative r and more formulae and programs.
%Y A098495 Rows include (-1)^c times A005408, A002878, A001834, A030221, A002315. Columns include A028387. Antidiagonal sums are in A098496.
%Y A098495 Adjacent sequences: A098492 A098493 A098494 this_sequence A098496 A098497 A098498
%Y A098495 Sequence in context: A125692 A128258 A104967 this_sequence A095025 A069897 A107682
%K A098495 sign,tabl
%O A098495 0,8
%A A098495 Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 12 2004
%E A098495 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 10 2005

%I A095025
%S A095025 1,1,2,1,1,1,3,1,1,1,1,1,2,0,2,1,0,1,2,0,1,1,1,1,0,2,1,1,3,1,3,0,1,0,0,
%T A095025 1,1,4,1,1,0,1,0,0,0,1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,6,0,2,0,0,1,1,0,1,1,
%U A095025 1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,1,1,0
%N A095025 Number of cyclic difference sets with n elements.
%C A095025 A (v,k,lambda) cyclic difference set is a subset D={d_1,d_2,...,d_k} of the integers modulo v such that {1,2,...,v-1} can each be represented as a difference (d_i-d_j) modulo v in exactly lambda different ways.
%H A095025 Dan Gordon, <a href="http://www.ccrwest.org/diffsets/diff_sets/">La Jolla Difference Set Repository</a>
%H A095025 Len Baumert and Dan Gordon, <a href="http://www.ccrwest.org/diffsets/papers/">Papers on Difference Sets</a>
%H A095025 Dan Gordon, <a href="http://www.ccrwest.org/diffsets/ds_list.pdf">List of Cyclic Difference Sets</a>
%e A095025 a(3)=1 corresponds to the (7,3,1) set {1,2,4}, a(4)=1 corresponds to the (14,4,1) set {0,1,3,9}.
%e A095025 a(5)=2 because there are two cyclic difference sets of length 5: The (v,k,lambda)=(11,5,2) set A095028={1,3,4,5,9} and the (21,5,1) set A095029= {3,6,7,12,14}
%Y A095025 Cf. A095029-A095047 examples of cyclic difference set with k=5..20.
%Y A095025 Adjacent sequences: A095022 A095023 A095024 this_sequence A095026 A095027 A095028
%Y A095025 Sequence in context: A128258 A104967 A098495 this_sequence A069897 A107682 A085476
%K A095025 nonn
%O A095025 3,3
%A A095025 Hugo Pfoertner (hugo(AT)pfoertner.org), May 27 2004

%I A069897
%S A069897 1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,1,2,2,5,1,1,1,6,1,3,1,2,1,1,3,8,1,1,
%T A069897 1,9,4,2,1,3,1,5,1,11,1,1,1,2,5,6,1,1,2,3,6,14,1,2,1,15,2,1,2,5,1,8,7,
%U A069897 3,1,1,1,18,1,9,1,6,1,2,1,20,1,3,3,21,9,5,1,2,1,11,10,23,3,1,1,3,3,2,1
%N A069897 Integer quotient of largest and smallest prime factors of n.
%F A069897 a(n)=Floor[A006530(n)/A020639(n)]
%Y A069897 Cf. A006530, A020639, A046665.
%Y A069897 Adjacent sequences: A069894 A069895 A069896 this_sequence A069898 A069899 A069900
%Y A069897 Sequence in context: A104967 A098495 A095025 this_sequence A107682 A085476 A124944
%K A069897 nonn
%O A069897 2,9
%A A069897 Labos E. (labos(AT)ana.sote.hu), Apr 10 2002

%I A107682
%S A107682 0,1,1,2,1,1,1,3,1,1,1,1,4,1,1,1,1,1,5,1,1,1,1,1,1,6,1,1,1,1,1,1,1,7,1,
%T A107682 1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,9,9,2,2,2,3,3,2,2,2,2,4,4,2,2,2,2,2,
%U A107682 5,5,2,2,2,2,2,2,6,6,2,2,2,2,2,2,2,7,7,2,2,2,2,2,2,2,2,8,8,2,2,2,2,2,2
%N A107682 The size of every run of identical digits is given by the digit immediately following immediately the said run. There can be no two identical runs in the sequence.
%C A107682 The sequence is finite.
%H A107682 E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/AutoChunks.htm">Auto-chunks</a>.
%e A107682 First run ("0") is made of 1 "identical digit(s)", thus is followed by 1; second run ("11") is made of 2 identical digits, thus followed by 2; third run ("2") is made of 1 "identical digit(s)", thus followed by 1; etc.
%Y A107682 Adjacent sequences: A107679 A107680 A107681 this_sequence A107683 A107684 A107685
%Y A107682 Sequence in context: A098495 A095025 A069897 this_sequence A085476 A124944 A094392
%K A107682 base,easy,fini,nonn
%O A107682 0,4
%A A107682 Eric Angelini (eric.angelini(AT)kntv.be), Jun 10 2005

%I A085476
%S A085476 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,4,3,1,1,1,1,5,6,1,2,1,1,1,6,10,4,1,1,1,
%T A085476 1,1,7,15,10,1,3,1,1,1,1,8,21,20,5,1,3,2,1,1,1,9,28,35,15,1,4,1,1,1,1,1,
%U A085476 10,36,56,35,6,1,6,1,1,1,1
%N A085476 Periodic Pascal array, read by antidiagonals.
%C A085476 G.f. of binomial transform of k-th row is given by 1/((1-x)^(k+1)-x^(k+1))
%F A085476 Square array T(n, k)=C(k, n mod (k+1)).
%e A085476 Rows begin
%e A085476 1 1 1 1 1 1 ...
%e A085476 1 1 1 1 1 1 ...
%e A085476 1 2 1 1 2 1 ...
%e A085476 1 3 3 1 1 3 ...
%e A085476 1 4 6 4 1 1 ...
%Y A085476 Cf. A007318, A000749, A049016, A006090, A049017.
%Y A085476 Adjacent sequences: A085473 A085474 A085475 this_sequence A085477 A085478 A085479
%Y A085476 Sequence in context: A095025 A069897 A107682 this_sequence A124944 A094392 A111946
%K A085476 easy,nonn,tabl
%O A085476 0,8
%A A085476 Paul Barry (pbarry(AT)wit.ie), Jul 02 2003

%I A124944
%S A124944 1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,4,3,1,1,1,1,6,4,1,1,1,1,1,8,6,3,1,1,1,1,
%T A124944 1,11,8,5,1,1,1,1,1,1,15,11,7,3,1,1,1,1,1,1,20,15,9,5,1,1,1,1,1,1,1,26,
%U A124944 21,12,8,3,1,1,1,1,1,1,1,35,27,16,10,5,1,1,1,1,1,1,1,1,45,37,21,13,8,3
%N A124944 Table, number of partitions of n with k as high median.
%C A124944 For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
%e A124944 For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
%Y A124944 Cf. A124943, A026794, A008284, A000041 (row sums), A027336 (1st column).
%Y A124944 Adjacent sequences: A124941 A124942 A124943 this_sequence A124945 A124946 A124947
%Y A124944 Sequence in context: A069897 A107682 A085476 this_sequence A094392 A111946 A137844
%K A124944 nonn,tabl
%O A124944 1,7
%A A124944 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 13 2006

%I A094392
%S A094392 1,1,1,1,1,2,1,1,1,3,1,1,1,1,5,1,1,1,1,2,8,1,1,1,1,1,3,13,1,1,1,1,1,1,5,
%T A094392 21,1,1,1,1,1,1,2,7,34,1,1,1,1,1,1,1,3,11,55,1,1,1,1,1,1,1,1,5,16,891,1,
%U A094392 1,1,1,1,1,1,2,7,25,144,1,1,1,1,1,1,1,1,1,3,11,37,233,1,1,1,1,1,1,1,1,1
%N A094392 Antidiagonals of the tables formed from b(m,2,n,n), which is defined in the reference.
%C A094392 This sequence can be used to help find an extension for A006209.
%D A094392 B.-S. Du, A simple method which generates infinitely many congruence indentities, Fib. Quart., 27 (1989), 116-124.
%F A094392 For i=2 and k >= 1 b(k+2, 2, n, n)=b(k, 2, 1, n) + b(k+1, 2, n, n). The remaining portion for the recurrence is defined in the reference.
%e A094392 E.g. for m = 5 and n = 2, b(5,2,2,2)= b(3,2,1,2) + b(4,2,2,2)= 2 because of the definition in the reference.
%e A094392 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 5 2 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 8 3 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 13 5 2 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 21 7 3 1 1 1 1 1 1 1 1 1 1 1 1
%e A094392 34 11 5 2 1 1 1 1 1 1 1 1 1 1 1
%e A094392 55 16 7 3 1 1 1 1 1 1 1 1 1 1 1
%e A094392 89 25 11 5 2 1 1 1 1 1 1 1 1 1 1
%e A094392 144 37 15 7 3 1 1 1 1 1 1 1 1 1 1
%e A094392 233 57 23 11 5 2 1 1 1 1 1 1 1 1 1
%e A094392 377 85 32 15 7 3 1 1 1 1 1 1 1 1 1
%e A094392 610 130 49 23 11 5 2 1 1 1 1 1 1 1 1
%p A094392 b := proc(k,i,j,n) option remember; if k = 1 then if i = 1 then return 0; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if k = 2 then if i = 1 then return 1; end if; if i = 2 then if j = n then return 1; end if; return 0; end if; end if; if j = n then return b(k-2, i, 1, n) + b(k-1, i, n, n); end if; return b(k-2, i, 1, n) + b(k-2, i, j+1, n); end proc; (Deugau)
%Y A094392 Cf. A006209.
%Y A094392 Adjacent sequences: A094389 A094390 A094391 this_sequence A094393 A094394 A094395
%Y A094392 Sequence in context: A107682 A085476 A124944 this_sequence A111946 A137844 A079229
%K A094392 nonn,tabl
%O A094392 1,6
%A A094392 Amy Robinson (amylou(AT)mchsi.com), Apr 28 2004
%E A094392 Corrected and extended by Chris Deugau (deugaucj(AT)uvic.ca), Dec 19 2005

%I A111946
%S A111946 1,1,1,1,1,2,1,1,1,3,1,1,1,1,5,1,1,2,1,1,8,1,1,1,1,1,1,13,1,1,1,3,1,1,1,
%T A111946 21,1,1,2,1,1,2,1,1,34,1,1,1,1,5,1,1,1,1,55,1,1,1,1,1,1,1,1,1,1,89,1,1,
%U A111946 2,3,1,8,1,3,2,1,1,144,1,1,1,1,1,1,1,1,1,1,1,1,233,1,1,1,1,1,1,13,1,1,1
%N A111946 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.
%D A111946 P. Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Q. 43 (No. 1, 2005), 3-14.
%F A111946 T(n, k) = Fibonacci(g), where g = gcd(n, k).
%Y A111946 Cf. A000045, A111956, A111957.
%Y A111946 Adjacent sequences: A111943 A111944 A111945 this_sequence A111947 A111948 A111949
%Y A111946 Sequence in context: A085476 A124944 A094392 this_sequence A137844 A079229 A115621
%K A111946 nonn,tabl
%O A111946 1,6
%A A111946 njas, Nov 28 2005

%I A137844
%S A137844 1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,5,1,1,1,2,
%T A137844 1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,6,1,1,1,2,1,1,1,3,
%U A137844 1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,5,1,1,1,2,1,1,1,3,1,1,1,2
%N A137844 Define S(1) = {1}, S(n+1) = S(n) U S(n) if a(n) is even, S(n+1) = S(n) U n U S(n) if a(n) is odd. Sequence {a(n), n >= 1} is limit as n approaches infinity of S(n). (U represents concatenation.).
%e A137844 S(1) = {1}.
%e A137844 S(2) = {1,1,1}, because a(1) = 1, which is odd.
%e A137844 S(3) = {1,1,1,2,1,1,1}, because a(2) = 1, which is odd.
%e A137844 S(4) = {1,1,1,2,1,1,1,3,1,1,1,2,1,1,1}.
%e A137844 S(5) = {1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1}, because a(4) = 2, which is even.
%e A137844 Etc.
%Y A137844 Cf. A137843.
%Y A137844 Adjacent sequences: A137841 A137842 A137843 this_sequence A137845 A137846 A137847
%Y A137844 Sequence in context: A124944 A094392 A111946 this_sequence A079229 A115621 A077565
%K A137844 easy,nonn
%O A137844 1,4
%A A137844 Leroy Quet (qq-quet(AT)mindspring.com), Feb 13 2008

%I A079229
%S A079229 1,1,2,1,1,1,3,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,2,1,3,1,1,1,1,2,1,1,1,2,
%T A079229 1,1,1,2,1,1,1,3,2,1,1,4,1,1,1,2,1,2,1,2,1,1,1,2,1,1,3,2,1,1,1,2,1,1,1,
%U A079229 2,1,1,3,1,1,1,1,3,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,4,1,2,1,1,1,2,1,1
%N A079229 Least k>0 such that sfk(n+k) > sfk(n), where sfk is the square-free kernel (A007947).
%Y A079229 a(n) = A079228(n) - n.
%Y A079229 Adjacent sequences: A079226 A079227 A079228 this_sequence A079230 A079231 A079232
%Y A079229 Sequence in context: A094392 A111946 A137844 this_sequence A115621 A077565 A115561
%K A079229 nonn
%O A079229 1,3
%A A079229 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 02 2003

%I A115621
%S A115621 1,1,2,1,1,1,3,1,1,1,2,1,2,4,1,1,1,1,1,1,2,1,2,1,3,5,1,1,1,1,1,2,1,2,1,
%T A115621 1,1,3,1,3,2,2,1,4,6,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,3,1,1,2,1,3,1,4,
%U A115621 2,3,1,5,7,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,2,1,3,1,1,2,2,2,1,1,2
%N A115621 Signature of partitions in Abramowitz and Stegun order.
%C A115621 The signature of a partition is a partition consisting of the repetition factors of the original partition. E.g. [1,1,3,4,4] = [1^2,3^1,4^2], so the repetition factors are 2,1,2, making the signature [1,2,2] = [1,2^2].
%C A115621 The sum (or order) of the signature is the number of parts of the original partition, and the number of parts of the signature is the number of distinct parts of the original partition.
%H A115621 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%e A115621 [1]; [1], [2]; [1], [1,1], [3]; [1], [1,1], [2], [1,2], [4]; ...
%Y A115621 Cf. A036036, A113787, A115622, part counts A103921, row counts A000070.
%Y A115621 Adjacent sequences: A115618 A115619 A115620 this_sequence A115622 A115623 A115624
%Y A115621 Sequence in context: A111946 A137844 A079229 this_sequence A077565 A115561 A115622
%K A115621 nonn,tabf
%O A115621 1,3
%A A115621 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 25 2006

%I A077565
%S A077565 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,3,1,3,1,1,1,4,1,1,2,3,1,4,1,3,1,1,1,
%T A077565 3,1,1,1,4,1,4,1,3,3,1,1,6,1,3,1,3,1,4,1,4,1,1,1,5,1,1,3,4,1,4,1,3,1,4,
%U A077565 1,7,1,1,3,3,1,4,1,6,2,1,1,5,1,1,1,4,1,5,1,3,1,1,1,9,1,3,3,3,1,4,1,4,4
%N A077565 Number of factorizations ( Smarandache Factor Partitions ) into parts with distinct prime signature.
%D A077565 Amarnath Murthy, Generalization of partition function. Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 1-2-3,2000.
%e A077565 a(24) = 4, 24 = 12*2 = 8*3 = 6*4. The partitions 2*3*4, 2*2*2*3 etc. are not counted.
%Y A077565 Cf. A077563, A077564, A077566.
%Y A077565 Adjacent sequences: A077562 A077563 A077564 this_sequence A077566 A077567 A077568
%Y A077565 Sequence in context: A137844 A079229 A115621 this_sequence A115561 A115622 A108886
%K A077565 nonn
%O A077565 1,8
%A A077565 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 11 2002
%E A077565 Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Aug 26 2003

%I A115561
%S A115561 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,3,1,5,1,1,1,2,1,1,3,7,1,5,1,2,1,1,1,
%T A115561 3,1,1,1,2,1,7,1,11,5,1,1,2,1,5,1,13,1,3,1,2,1,1,1,3,1,1,7,2,1,11,1,17,
%U A115561 1,7,1,2,1,1,5,19,1,13,1,2,3,1,1,3,1,1,1,2,1,3,1,23,1,1,1,2,1,7,11,5,1
%N A115561 lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.
%C A115561 a(n) = A020639(A054576(n)).
%H A115561 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeastPrimeFactor.html">Least Prime Factor</a>
%Y A115561 Cf. A032742, A014673, A117358.
%Y A115561 Adjacent sequences: A115558 A115559 A115560 this_sequence A115562 A115563 A115564
%Y A115561 Sequence in context: A079229 A115621 A077565 this_sequence A115622 A108886 A001492
%K A115561 nonn
%O A115561 1,8
%A A115561 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 10 2006

%I A115622
%S A115622 1,1,2,1,1,1,3,1,1,1,2,2,1,4,1,1,1,1,1,2,1,2,1,3,1,5,1,1,1,1,1,2,1,2,1,
%T A115622 1,1,3,1,3,2,2,4,1,6,1,1,1,1,1,2,1,1,1,1,1,1,3,1,2,1,2,1,2,1,1,4,1,3,1,
%U A115622 3,2,5,1,7,1,1,1,1,1,2,1,1,1,1,1,1,3,1,2,1,1,1,2,1,2,1,1,4,1,2,1,2,2,2
%N A115622 Signature of partitions in Mathematica order.
%C A115622 The signature of a partition is a partition consisting of the repetition factors of the original partition. E.g. [4,4,3,1,1] = [4^2,3^1,1^2], so the repetition factors are 2,1,2, making the signature [2,2,1] = [2^2,1].
%C A115622 The sum (or order) of the signature is the number of parts of the original partition, and the number of parts of the signature is the number of distinct parts of the original partition.
%e A115622 [1]; [1], [2]; [1], [1,1], [3]; [1], [1,1], [2], [2,1], [4]; ...
%Y A115622 Cf. A080577, A115624, A115621, part counts A115623, row counts A000070.
%Y A115622 Adjacent sequences: A115619 A115620 A115621 this_sequence A115623 A115624 A115625
%Y A115622 Sequence in context: A115621 A077565 A115561 this_sequence A108886 A001492 A054576
%K A115622 nonn,tabf
%O A115622 1,3
%A A115622 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 25 2006

%I A108886
%S A108886 1,1,1,1,2,1,1,1,3,1,1,1,3,4,1,1,1,5,2,5,1,1,1,1,20,5,6,1,1,1,1,35,35,3,
%T A108886 7,1,1,1,1,1,1,28,7,8,1,1,1,1,1,9,63,42,4,9,1,1,1,1,1,15,42,35,120,9,10,
%U A108886 1,1,1,1,1,1,231,11,66,55,5,11,1,1,1,1,1,1,396,231,72,45,55,11,12,1,1,1
%N A108886 Let T(m,p) be the value of the following game: there are m ``minus'' balls and p ``plus'' balls in an urn, for a total of n=m+p balls. You may draw balls from the urn one at a time at random and without replacement until you decide to stop drawing. Each minus ball drawn costs you $1 and each plus ball drawn gets you $1. Sequence gives triangle of denominators of T(n-p,p), 0 <= p <= n, read by rows.
%D A108886 L. A. Shepp, Stochastic Processes [Course], Statistics Dept., Rutgers University, 2004.
%F A108886 T(m, 0)=0, T(0, p)=p; T(m, p) = max{0, (m/(m+p))*(-1+T(m-1, p))+(p/(m+p))*(1+T(m, p-1))}.
%e A108886 Triangle of values T(n-p,p), 0 <= p <= n, begins:
%e A108886 [0]
%e A108886 [0, 1]
%e A108886 [0, 1/2, 2]
%e A108886 [0, 0, 4/3, 3]
%e A108886 [0, 0, 2/3, 9/4, 4]
%e A108886 [0, 0, 1/5, 3/2, 16/5, 5]
%e A108886 [0, 0, 0, 17/20, 12/5, 25/6, 6]
%e A108886 [0, 0, 0, 12/35, 58/35, 10/3, 36/7, 7]
%e A108886 [0, 0, 0, 0, 1, 71/28, 30/7, 49/8, 8]
%p A108886 M:=60; for m from 0 to M do T(m,0):=0; od: for p from 0 to M do T(0,p):=p; od: for n from 1 to M do for m from 1 to n-1 do p:=n-m; t1:=(m/(m+p))*(-1+T(m-1,p))+(p/(m+p))*(1+T(m,p-1)); T(m,p):=max(0,t1); od: od:
%Y A108886 Cf. A108885. Sequence T(m,m) is A108883/A108884.
%Y A108886 Adjacent sequences: A108883 A108884 A108885 this_sequence A108887 A108888 A108889
%Y A108886 Sequence in context: A077565 A115561 A115622 this_sequence A001492 A054576 A138904
%K A108886 nonn,tabl,frac
%O A108886 0,5
%A A108886 njas, Jul 16 2005

%I A001492
%S A001492 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,4,1,1,1,5,1,1,1,6,1,1,1,7,1,1,1,8,
%T A001492 1,1,1,9,1,1,1,10,1,1,1,11,1,1,1,12,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,4,
%U A001492 1,1,1,5,1,1,1,6,1,1,1,7,1,1,1,8,1,1,1,9
%N A001492 Clock chimes with a quarter-hour bell.
%o A001492 (PARI) a(n)=if((n+1)%4,1,((n+1)/4-1)%12+1)
%Y A001492 Cf. A007879, A007884.
%Y A001492 Adjacent sequences: A001489 A001490 A001491 this_sequence A001493 A001494 A001495
%Y A001492 Sequence in context: A115561 A115622 A108886 this_sequence A054576 A138904 A135222
%K A001492 nonn,easy
%O A001492 0,8
%A A001492 njas

%I A054576
%S A054576 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,4,1,3,1,5,1,1,1,6,1,1,3,7,1,5,1,8,1,1,1,
%T A054576 9,1,1,1,10,1,7,1,11,5,1,1,12,1,5,1,13,1,9,1,14,1,1,1,15,1,1,7,16,1,11,
%U A054576 1,17,1,7,1,18,1,1,5,19,1,13,1,20,9,1,1,21,1,1,1,22,1,15,1,23,1,1,1,24
%N A054576 Largest proper factor of largest proper factor of n.
%F A054576 a(n) = A053598(A053598(n))
%F A054576 a(n)=A032742(A032742(n)); A117357(n)=A020639(a(n)); A117358(n)=A032742(a(n))=a(n)/A117357(n); a(A037143(n))=1, a(A033942(n))>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 10 2006
%Y A054576 Cf. A053598.
%Y A054576 Adjacent sequences: A054573 A054574 A054575 this_sequence A054577 A054578 A054579
%Y A054576 Sequence in context: A115622 A108886 A001492 this_sequence A138904 A135222 A124094
%K A054576 easy,nonn
%O A054576 1,8
%A A054576 Henry Bottomley (se16(AT)btinternet.com), Apr 11 2000

%I A138904
%S A138904 1,1,1,2,1,1,1,3,1,1,2,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,
%T A138904 1,2,1,1,1,1,1,3,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1,
%U A138904 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A138904 Number of rotational symmetries in the binary expansion of a number.
%C A138904 Mersenne numbers of form (2^n - 1) have n rotational symmetries.
%C A138904 For prime length binary expansions these are the only nontrivial symmetries.
%C A138904 For composite length expansions it seems that when the number of symmetries is non-trivial it is equal to a factor of the length. We're working on an explicit formula.
%C A138904 Discovered in the context of random circulant matricies, examining if there's a correlation between degrees of freedom and number of symmetries in the first row.
%C A138904 When combined with A138954, these two sequences should give a full account of the number of redundant rows in a circulant square matrix with at most two distinct values, where a(n) is the encoding of the first row of the matrix into binary such that value a = 1, and value b = 0.
%C A138904 Discovered on the night of Apr 02, 2008 by Maxwell Sills and Gary Doran.
%C A138904 Conjecture: For binary expansions of length n, there are d(n) distinct values that will show up as symmetries, where d is the divisor function. The symmetry values will be precisely the divisors of n.
%C A138904 Example: for binary expansions of length 12, one sees that d(12) = 6 distinct values show up as symmetries (1, 2, 3, 4, 6, 12).
%C A138904 Conjecture: For numbers whose binary expansion has length n which has proper divisors which are all coprime: There will be only one number of length n with n symmetires. That number is 2^n - 1. For each proper divisor d (excluding 1) , you can generate all numbers of length n that have n/d symmetries like so: (2^0 + 2^d + 2^2d ... 2^(n-d)) * a, where 2^(d-1) <= a < (2^d) - 1. The rest of the expansions of length n will have only the trivial symmetry.
%H A138904 Maxwell Sills and Gary Doran, <a href="http://www.research.att.com/~njas/sequences/b138904.txt">Table of n, a(n) for n = 0..99</a>
%e A138904 a(10) = 2 because the binary expansion of 10 is 1010 and it has two rotational symmetries (including identity).
%Y A138904 Cf. A136441, A138954.
%Y A138904 Adjacent sequences: A138901 A138902 A138903 this_sequence A138905 A138906 A138907
%Y A138904 Sequence in context: A108886 A001492 A054576 this_sequence A135222 A124094 A101950
%K A138904 base,easy,nonn
%O A138904 0,4
%A A138904 Max Sills (maxwell.sills(AT)case.edu), Apr 03 2008, Apr 04 2008

%I A135222
%S A135222 1,1,1,2,1,1,1,3,1,1,2,1,4,1,1,1,4,1,5,1,1,2,1,7,1,6,1,1,1,5,1,11,1,7,1,
%T A135222 1,2,1,11,1,16,1,8,1,1,1,6,1,21,1,22,1,9,1,1
%N A135222 A049310 + A000012 - I.
%C A135222 Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28,...).
%F A135222 A049310 + A000012 - Identity matrix, as infinite lower triangular matrices.
%e A135222 First few rows of the triangle are:
%e A135222 1;
%e A135222 1, 1;
%e A135222 2, 1, 1;
%e A135222 1, 3, 1, 1;
%e A135222 2, 1, 4, 1, 1;
%e A135222 1, 4, 1, 5, 1, 1;
%e A135222 2, 1, 7, 1, 6, 1, 1;
%e A135222 1, 5, 1, 11, 1, 7, 1, 1;
%e A135222 2, 1, 11, 1, 16, 1, 8, 1, 1;
%e A135222 ...
%Y A135222 Cf. A049310, A081659.
%Y A135222 Adjacent sequences: A135219 A135220 A135221 this_sequence A135223 A135224 A135225
%Y A135222 Sequence in context: A001492 A054576 A138904 this_sequence A124094 A101950 A104562
%K A135222 nonn,tabl
%O A135222 1,4
%A A135222 Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007

%I A124094
%S A124094 1,1,1,1,1,2,1,1,1,3,1,1,2,2,5,1,1,1,2,2,7,1,1,2,2,4,3,11,1,1,1,3,2,5,4,
%T A124094 15,1,1,2,1,5,3,7,5,22,1,1,1,3,1,6,4,9,6,30,1,1,2,2,5,2,10,5,13,8,42,1,
%U A124094 1,1,2,2,7,2,13,6,16,10,56,1,1,2,2,4,3,11,3,19,8,22,12,77,1,1,1,3,2,5,4
%N A124094 Table T(n,m) giving number of partitions of n such that all parts are coprime to m. Read along antidiagonals (increasing n, decreasing m).
%H A124094 N. Robbins, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Robbins/robbins4.html">On partition functions and divisor sums</a>, J. Int. Sequences, 5 (2002) 02.1.4.
%e A124094 Upper left corner of table starts with row m=1 and column n=0:
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243,297,355,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,3,5,6,10,13,19,25,34,44,60, 76,100,127,164,205,262,325,409,505,628,769,
%e A124094 1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23,26,
%e A124094 1,1,2,3,5,7,11,14,21,28,39,51,70, 90,119,153,199,252,324,406,515,642,804,994,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243,297,355,
%e A124094 1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41, 47,54,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,55,76, 99,132,171,224,286,370,468,597,750,945,1177,
%e A124094 1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23,26,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,100,134,174,228,292,378,479,612,770,972,1213,
%e A124094 1,1,1,2,2,3, 4, 4, 5, 7, 8,10,12, 14, 17, 21, 24, 28, 34, 39, 46, 53, 61,71,
%e A124094 1,1,2,2,4,4, 6, 7,11,12,16,19,25, 29, 37, 44, 56, 65, 80, 94,114,133,160,187,
%e A124094 1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89,104,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,296,384,488,624,787,995,1244,
%e A124094 1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23,26,
%e A124094 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,489,626,790,999,1250,
%e A124094 1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41, 47,54,
%o A124094 (PARI) sigmastar(n,m)= { local(d,res=0) ; d=divisors(n) ; for(i=1,matsize(d)[2], if( gcd(d[i],m)==1, res += d[i] ; ) ; ) ; return(res) ; } f(n,m)= { local(qvec=vector(n+1,i,gcd(1,m))) ; for(i=1,n, qvec[i+1]=sum(k=1,i,sigmastar(k,m)*qvec[i-k+1])/i ; ) ; return(qvec[n+1]) ; } { for(d=1,18, for(c=0,d-1, r=d-c ; print1(f(c,r),",") ; ) ; ) ; }
%Y A124094 Row m=1 is A000041. Rows m=2,4,8,... (where m is a power of 2) are A000009. Rows m=3,9,... (where m is a power of 3) are A000726. Row m=5 is A035959. Row=7 is A035985. Row m=10 is A096938.
%Y A124094 Adjacent sequences: A124091 A124092 A124093 this_sequence A124095 A124096 A124097
%Y A124094 Sequence in context: A054576 A138904 A135222 this_sequence A101950 A104562 A111603
%K A124094 easy,nonn,tabl
%O A124094 0,6
%A A124094 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 26 2006

%I A101950
%S A101950 1,1,1,0,2,1,1,1,3,1,1,2,3,4,1,0,4,2,6,5,1,1,2,9,0,10,6,1,1,3,9,15,5,15,
%T A101950 7,1,0,6,3,24,20,14,21,8,1,1,3,18,6,49,21,28,28,9,1,1,4,18,36,35,84,14,
%U A101950 48,36,10,1,0,8,4,60,50,98,126,6,75,45,11,1,1,4,30,20,145,36,210
%V A101950 1,1,1,0,2,1,-1,1,3,1,-1,-2,3,4,1,0,-4,-2,6,5,1,1,-2,-9,0,10,6,1,1,3,-9,-15,5,15,7,1,0,
%W A101950 6,3,-24,-20,14,21,8,1,-1,3,18,-6,-49,-21,28,28,9,1,-1,-4,18,36,-35,-84,-14,48,36,10,1,
%X A101950 0,-8,-4,60,50,-98,-126,6,75,45,11,1,1,-4,-30,20,145,36,-210
%N A101950 Product of A049310 and A007318 as lower triangular matrices.
%C A101950 A Chebyshev and Pascal product.
%C A101950 Row sums are n+1, diagonal sums the constant sequence 1. Riordan array (1/(1-x+x^2,x/(1-x+x^2)).
%C A101950 Apart from signs, identical with A104562.
%F A101950 Number triangle T(n, k) = sum{k=0..n, (-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}
%e A101950 Rows begin {1}, {1,1}, {0,2,1}, {-1,1,3,1}, { -1,-2,3,4,1},..
%Y A101950 Cf. A104562.
%Y A101950 Adjacent sequences: A101947 A101948 A101949 this_sequence A101951 A101952 A101953
%Y A101950 Sequence in context: A138904 A135222 A124094 this_sequence A104562 A111603 A136178
%K A101950 easy,sign,tabl
%O A101950 0,5
%A A101950 Paul Barry (pbarry(AT)wit.ie), Dec 22 2004

%I A104562
%S A104562 1,1,1,0,2,1,1,1,3,1,1,2,3,4,1,0,4,2,6,5,1,1,2,9,0,10,6,1,1,3,9,15,5,15,7,
%T A104562 1,0,6,3,24,20,14,21,8,1,1,3,18,6,49,21,28,28,9,1,1,4,18,36,35,84,14,48,
%U A104562 36,10,1,0,8,4,60,50,98,126,6,75,45,11,1,1,4,30,20,145,36,210,168,45,110,55
%V A104562 1,-1,1,0,-2,1,1,1,-3,1,-1,2,3,-4,1,0,-4,2,6,-5,1,1,2,-9,0,10,-6,1,-1,3,9,-15,-5,15,-7,
%W A104562 1,0,-6,3,24,-20,-14,21,-8,1,1,3,-18,-6,49,-21,-28,28,-9,1,-1,4,18,-36,-35,84,-14,-48,
%X A104562 36,-10,1,0,-8,4,60,-50,-98,126,6,-75,45,-11,1,1,4,-30,-20,145,-36,-210,168,45,-110,55
%N A104562 Inverse of the Motzkin triangle A064189.
%C A104562 Or, triangle read by rows: T(0,0)=1; for n>=1 T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the n X n tridiagonal matrix with 1's on the main, sub-, and superdiagonal (0<=k<=n). The characteristic polynomial has a root 1+2cos(Pi/(n+1)). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 19 2006
%C A104562 Row sums have g.f. 1/(1+x^2); diagonal sums are (-1)^n. Riordan array (1/(1+x+x^2), x/(1+x+x^2)).
%C A104562 Apart from signs, identical to A101950.
%C A104562 Or, triangle read by rows in which row n gives coefficients of characteristic polynomial of tridaigonal matrix with 1's on the main diagonal and -1's on the two adjacent diagonals. For example: M(3)={{1, -1, 0}, {-1, 1, -1}, {0, -1, 1}}. - from More terms from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 15 2008
%D A104562 Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
%F A104562 Number triangle T(n, k)=sum{j=0..n, (-1)^(k-j)*(-1)^((n-j)/2) C((n+j)/2, j)(1+(-1)^(n+j))C(j, k)/2}
%e A104562 Triangle starts:
%e A104562 1;
%e A104562 -1,1;
%e A104562 0,-2,1;
%e A104562 1,1,-3,1;
%e A104562 -1,2,3,-4,1;
%e A104562 0,-4,2,6,-5,1;
%p A104562 with(linalg): m:=proc(i,j) if abs(i-j)<=1 then 1 else 0 fi end: T:=(n,k)->coeff(charpoly(matrix(n,n,m),x),x,k): 1; for n from 1 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A104562 a0[n_] := 1; b[n_] := -1; T[n_, m_, d_] := If[ n == m, a0[n], If[n == m - 1 || n == m + 1, If[n == m - 1, b[m - 1], If[n == m + 1, b[n - 1], 0]], 0]]; MO[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ MO[n], x], x], {n, 1, 10}]]; Flatten[a] - from More terms from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 15 2008
%Y A104562 Cf. A125090, A101950.
%Y A104562 Adjacent sequences: A104559 A104560 A104561 this_sequence A104563 A104564 A104565
%Y A104562 Sequence in context: A135222 A124094 A101950 this_sequence A111603 A136178 A047140
%K A104562 easy,sign,tabl
%O A104562 0,5
%A A104562 Paul Barry (pbarry(AT)wit.ie), Mar 15 2005
%E A104562 Edited by njas, Apr 10 2008

%I A111603
%S A111603 1,1,1,1,2,1,1,1,3,1,1,2,3,4,1,1,2,1,2,5,1,1,2,3,4,5,6,1,1,1,3,3,5,3,7,
%T A111603 1,1,2,3,4,5,2,7,8,1,1,2,3,4,1,3,7,4,9,1,1,2,3,4,5,6,7,8,9,10,1,1,1,3,1,
%U A111603 5,6,7,2,3,5,11,1,1,2,3,4,5,6,7,8,9,10,11,12,1,1,1,3,4,5,3,1,4,9,10
%N A111603 Consider the array T(n, m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from upper right to lower left.
%e A111603 Table begins
%e A111603 k= 0 1 2 3 4 5 6 7 8 9 10 11 12 13
%e A111603 n\
%e A111603 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A111603 2| 1 2 1 2 2 2 1 2 2 2 1 2 1 2
%e A111603 3| 1 3 3 1 3 3 3 3 3 3 3 3 1 3
%e A111603 4| 1 4 2 4 3 4 4 4 1 4 4 4 3 4
%e A111603 5| 1 5 5 5 5 1 5 5 5 5 4 5 5 5
%e A111603 6| 1 6 3 2 3 6 6 6 3 4 6 6 6 6
%e A111603 7| 1 7 7 7 7 7 7 1 7 7 7 7 7 7
%e A111603 8| 1 8 4 8 2 8 4 8 7 8 8 8 4 8
%e A111603 9| 1 9 9 3 9 9 3 9 9 1 9 9 6 9
%e A111603 10| 1 10 5 10 10 2 5 10 10 10 3 10 5 10
%e A111603 11| 1 11 11 11 11 11 11 11 11 11 11 1 11 11
%e A111603 12| 1 12 6 4 9 12 4 12 12 8 6 12 6 12
%e A111603 13| 1 13 13 13 13 13 13 13 13 13 13 13 13 1
%e A111603 14| 1 14 7 14 7 14 14 2 7 14 14 14 14 14
%e A111603 15| 1 15 15 5 15 3 10 15 15 10 15 15 5 15
%e A111603 16| 1 16 8 16 4 16 8 16 10 16 8 16 12 16
%t A111603 f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[ a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]]; Flatten[ Table[ f[i, n - i], {n, 15}, {i, n - 1, 1, -1}]]
%Y A111603 Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
%Y A111603 Cf. A109626, A111604.
%Y A111603 Adjacent sequences: A111600 A111601 A111602 this_sequence A111604 A111605 A111606
%Y A111603 Sequence in context: A124094 A101950 A104562 this_sequence A136178 A047140 A047150
%K A111603 nonn,tabl
%O A111603 1,5
%A A111603 Paul D. Hanna (pauldhanna(AT)juno.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 01 2005

%I A136178
%S A136178 1,1,1,2,1,1,1,3,1,1,2,4,1,3,1,1,5,1,1,1,1,2,6,1,1,1,7,1,1,5,1,2,4,8,1,
%T A136178 1,1,3,3,9,1,1,2,1,5,10
%N A136178 Irregular array read by rows: row n contains the GCDs of each pair of consecutive positive divisors of n.
%C A136178 Each row has d(n)-1 terms, where d(n) is the number of positive divisors of n. The first row listed is row 2.
%e A136178 The positive divisors of 20 are 1,2,4,5,10,20. GCD(1,2)=1. GCD(2,4)=2. GCD(4,5)=1. GCD(5,10)=5. And GCD(10,20)=10. So row 20 is (1,2,1,5,10).
%Y A136178 Cf. A136179, A136180, A136181.
%Y A136178 Adjacent sequences: A136175 A136176 A136177 this_sequence A136179 A136180 A136181
%Y A136178 Sequence in context: A101950 A104562 A111603 this_sequence A047140 A047150 A102054
%K A136178 more,nonn,tabl
%O A136178 2,4
%A A136178 Leroy Quet (qq-quet(AT)mindspring.com), Dec 19 2007

%I A047140
%S A047140 1,1,1,1,2,1,1,1,3,1,1,2,4,4,1,1,3,2,8,5,1,1,4,5,10,13,6,1,1,5,9,15,23,
%T A047140 19,7,1,1,6,14,24,38,42,26,8,1,1,7,20,38,62,80,68,34,9,1,1,8,27,58,38,
%U A047140 142,148,102,43,10,1,1,9,35,85,96,180,290
%N A047140 Array read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right, and no up-step crosses the line y=4x/3. (Thus a path crosses the line only at lattice points and on right-steps.).
%Y A047140 Adjacent sequences: A047137 A047138 A047139 this_sequence A047141 A047142 A047143
%Y A047140 Sequence in context: A104562 A111603 A136178 this_sequence A047150 A102054 A111604
%K A047140 nonn,tabl
%O A047140 0,5
%A A047140 Clark Kimberling (ck6(AT)evansville.edu). Definition revised Dec 08 2006

%I A047150
%S A047150 1,1,1,1,2,1,1,1,3,1,1,2,4,4,1,1,3,6,8,5,1,1,4,3,14,13,6,1,1,5,7,17,27,
%T A047150 19,7,1,1,6,12,24,44,46,26,8,1,1,7,18,36,68,90,72,34,9,1,1,8,25,54,104,
%U A047150 158,162,106,43,10,1,1,9,33,79,54,262,320
%N A047150 Array read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right, and no up-step crosses the line y=5x/3. (Thus a path crosses the line only at lattice points and on right-steps.).
%Y A047150 Adjacent sequences: A047147 A047148 A047149 this_sequence A047151 A047152 A047153
%Y A047150 Sequence in context: A111603 A136178 A047140 this_sequence A102054 A111604 A101491
%K A047150 nonn,tabl
%O A047150 0,5
%A A047150 Clark Kimberling (ck6(AT)evansville.edu). Definition revised Dec 08 2006

%I A102054
%S A102054 1,1,1,1,2,1,1,1,3,1,1,4,2,4,1,1,13,26,10,5,1,1,142,229,116,25,6,1,1,1931,3181,
%T A102054 1567,371,49,7,1,1,36296,59700,29464,6922,952,84,8,1,1,893273,1469380,725108,
%U A102054 170398,23358,2100,132,9,1,1,27927346,45938639,22669816,5327198,730252,65526,4152
%V A102054 1,1,1,1,2,1,1,1,3,1,1,4,-2,4,1,1,-13,26,-10,5,1,1,142,-229,116,-25,6,1,1,-1931,3181,
%W A102054 -1567,371,-49,7,1,1,36296,-59700,29464,-6922,952,-84,8,1,1,-893273,1469380,-725108,
%X A102054 170398,-23358,2100,-132,9,1,1,27927346,-45938639,22669816,-5327198,730252,-65526,4152
%N A102054 Triangular matrix, read by rows, where T(n,k) = T(n-1,k) - [T^-1](n-1,k-1); also equals the matrix inverse of A060083 (Euler polynomials).
%C A102054 Column 1 forms A102055. Column 2 forms A102056.
%F A102054 T(n, k) = T(n-1, k) - A060083(n-1, k-1), for n>0, with T(0, 0)=1.
%e A102054 T(5,3) = -10 = T(4,3) - A060083(4,2) = 4 - 14.
%e A102054 T(6,2) = -229 = T(5,2) - A060083(5,1) = 26 - 255.
%e A102054 Rows begin:
%e A102054 [1],
%e A102054 [1,1],
%e A102054 [1,2,1],
%e A102054 [1,1,3,1],
%e A102054 [1,4,-2,4,1],
%e A102054 [1,-13,26,-10,5,1],
%e A102054 [1,142,-229,116,-25,6,1],
%e A102054 [1,-1931,3181,-1567,371,-49,7,1],
%e A102054 [1,36296,-59700,29464,-6922,952,-84,8,1],...
%e A102054 The matrix inverse is equal to A060083:
%e A102054 [1],
%e A102054 [ -1,1],
%e A102054 [1,-2,1],
%e A102054 [ -3,5,-3,1],
%e A102054 [17,-28,14,-4,1],
%e A102054 [ -155,255,-126,30,-5,1],...
%o A102054 (PARI) {T(n,k)=local(M=matrix(n+1,n+1));M[1,1]=1;if(n>0,M[2,1]=1;M[2,2]=1); for(r=3,n+1, for(c=1,r,M[r,c]=if(c==1,M[r-1,1], if(c==r,1,M[r,c]=M[r-1,c]-((matrix(r-1,r-1,i,j,M[i,j]))^-1)[r-1,c-1])))); return(M[n+1,k+1])}
%Y A102054 Cf. A060083, A102055, A102056.
%Y A102054 Adjacent sequences: A102051 A102052 A102053 this_sequence A102055 A102056 A102057
%Y A102054 Sequence in context: A136178 A047140 A047150 this_sequence A111604 A101491 A032436
%K A102054 sign,tabl
%O A102054 0,5
%A A102054 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 28 2004

%I A111604
%S A111604 1,1,1,1,2,1,1,1,3,1,1,4,3,2,1,1,2,1,2,5,1,1,6,5,4,3,2,1,1,1,3,3,5,3,7,
%T A111604 1,1,8,7,2,5,4,3,2,1,1,2,3,4,1,3,7,4,9,1,1,10,9,8,7,6,5,4,3,2,1,1,1,3,1,
%U A111604 5,6,7,2,3,5,11,1,1,12,11,10,9,8,7,6,5,4,3,2,1,1,1,3,4,5,3,1,4,9,10,11
%N A111604 Consider the array T(n, m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read zig-zag.
%C A111604 T(n,n)=T(n,n+2)=A111627.
%e A111604 Table begins
%e A111604 \k...0...1....2....3....4....5....6....7....8....9...10...11...12...13
%e A111604 n\
%e A111604 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A111604 2| 1 2 1 2 2 2 1 2 2 2 1 2 1 2
%e A111604 3| 1 3 3 1 3 3 3 3 3 3 3 3 1 3
%e A111604 4| 1 4 2 4 3 4 4 4 1 4 4 4 3 4
%e A111604 5| 1 5 5 5 5 1 5 5 5 5 4 5 5 5
%e A111604 6| 1 6 3 2 3 6 6 6 3 4 6 6 6 6
%e A111604 7| 1 7 7 7 7 7 7 1 7 7 7 7 7 7
%e A111604 8| 1 8 4 8 2 8 4 8 7 8 8 8 4 8
%e A111604 9| 1 9 9 3 9 9 3 9 9 1 9 9 6 9
%e A111604 10| 1 10 5 10 10 2 5 10 10 10 3 10 5 10
%e A111604 11| 1 11 11 11 11 11 11 11 11 11 11 1 11 11
%e A111604 12| 1 12 6 4 9 12 4 12 12 8 6 12 6 12
%e A111604 13| 1 13 13 13 13 13 13 13 13 13 13 13 13 1
%e A111604 14| 1 14 7 14 7 14 14 2 7 14 14 14 14 14
%e A111604 15| 1 15 15 5 15 3 10 15 15 10 15 15 5 15
%e A111604 16| 1 16 8 16 4 16 8 16 10 16 8 16 12 16
%t A111604 f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 32}]]; g[n_, m_] := f[n][[m]];
%Y A111604 Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
%Y A111604 Cf. A109626, A111603.
%Y A111604 Adjacent sequences: A111601 A111602 A111603 this_sequence A111605 A111606 A111607
%Y A111604 Sequence in context: A047140 A047150 A102054 this_sequence A101491 A032436 A073408
%K A111604 nonn,tabl
%O A111604 1,5
%A A111604 Paul D. Hanna (pauldhanna(AT)juno.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 01 2005

%I A101491
%S A101491 1,0,1,2,1,1,1,3,1,1,5,4,4,1,1,5,10,5,5,1,1,15,15,15,6,6,1,1,20,35,21,
%T A101491 21,7,7,1,1,50,56,56,28,28,8,8,1,1,76,126,84,84,36,36,9,9,1,1,176,210,
%U A101491 210,120,120,45,45,10,10,1,1,286,462,330,330,165,165,55,55,11,11,1,1
%N A101491 Triangle T(n,k), read by rows: number of Kn{\"o}del walks starting at 0, ending at k, with n steps.
%H A101491 H. Prodinger, <a href="http://www.mat.univie.ac.at/~slc/opapers/s50proding.html">The Kernel Method: a collection of examples</a>
%F A101491 G.f.: r(z)/[z(1+z)(1-r(z))]*(1+xzr(z))/(1-xr(z)), with r(z)=(1-sqrt(1-4z^2)/2z. Then the g.f. of the k-th column is r(z)^(k+1)/[z(1-r(z))].
%F A101491 T(n, k) = Sum{i=0..n, (-1)^(n-i)*C(i, [i/2]) } for k=0, otherwise T(n, k) = C(n, [(n-k)/2]).
%e A101491 1,
%e A101491 0,1,
%e A101491 2,1,1,
%e A101491 1,3,1,1,
%e A101491 5,4,4,1,1,
%e A101491 5,10,5,5,1,1,
%e A101491 15,15,15,6,6,1,1,
%e A101491 20,35,21,21,7,7,1,1,
%e A101491 50,56,56,28,28,8,8,1,1,
%e A101491 76,126,84,84,36,36,9,9,1,1,
%Y A101491 Left-hand columns include A086905, A037952, A037955, A037951, A037956, A037953, A037957, A037954, A037958.
%Y A101491 Adjacent sequences: A101488 A101489 A101490 this_sequence A101492 A101493 A101494
%Y A101491 Sequence in context: A047150 A102054 A111604 this_sequence A032436 A073408 A120454
%K A101491 nonn,tabl
%O A101491 0,4
%A A101491 Ralf Stephan, Jan 21 2005

%I A032436
%S A032436 1,1,1,2,1,1,1,3,1,2,1,2,4,1,1,3,1,2,5,3,1,2,1,1,2,6,1,4,3,3,1,1,2,7,3,
%T A032436 1,1,2,4,1,1,2,8,1,4,1,3,3,5,1,1,4,9,3,2,5,1,5,1,1,4,3,2,10,1,5,1,1,3,
%U A032436 8,2,1,1,1,2,11,3,1,5,6,4,2,4,3,1,1,1,7,12,5,2,3,2,1,9,4,5,7,1,1,6
%N A032436 Triangle of third-to-last man to survive in the Josephus problem of n men in a circle with every k-th killed, with k<=n.
%D A032436 Ball, W. W. R. and Coxeter, H. S. M., Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 32-36, 1987.
%D A032436 Kraitchik, M. "Josephus' Problem." Sec. 3.13 in Mathematical Recreations. New York: W.W. Norton, pp. 93-94, 1942.
%H A032436 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JosephusProblem.html">More information</a>
%Y A032436 Cf. A032434, A032435, A032436.
%Y A032436 Adjacent sequences: A032433 A032434 A032435 this_sequence A032437 A032438 A032439
%Y A032436 Sequence in context: A102054 A111604 A101491 this_sequence A073408 A120454 A076511
%K A032436 nonn,tabl
%O A032436 3,4
%A A032436 njas

%I A073408
%S A073408 1,1,1,1,2,1,1,1,3,1,2,1,3,2,1,1,4,1,3,2,4,1,2,1,4,1,3,1,5,1,1,2,5,2,4,
%T A073408 1,5,3,3,1,6,1,4,2,5,1,2,1,6,2,4,1,6,3,3,3,6,1,5,1,5,2,1,2,6,1,5,3,6,1,
%U A073408 4,1,6,3,5,2,7,1,3,1,7,1,6,4,6,2,4,1,7,2,5,3,6,2,2,1,6,4,6,1,7,1,4,2,7
%N A073408 Let cophi_m(x) denotes the cototient function applied m times to x (cophi(x)=x-phi(x)). Sequence gives the minimum number of iterations m such that cophi_m(n) divides n.
%F A073408 It seems that sum(k=1, n, a(k)) is asymptotic to C*n*log(n) with C<1
%e A073408 cophi(10) ->6 cophi(6)->4 cophi(4)->2 and 2 divides 10. Hence 3 iterations are needed and a(10)=3
%o A073408 (PARI) a(n)=if(n<0,0,c=1; s=n; while(n%(s-eulerphi(s))>0,s=s-eulerphi(s); c++); c)
%Y A073408 Cf. A019294.
%Y A073408 Adjacent sequences: A073405 A073406 A073407 this_sequence A073409 A073410 A073411
%Y A073408 Sequence in context: A111604 A101491 A032436 this_sequence A120454 A076511 A099244
%K A073408 easy,nonn
%O A073408 2,5
%A A073408 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 23 2002

%I A120454
%S A120454 1,1,1,1,1,2,1,1,1,3,1,2,1,4,2,1,1,2,1,3,3,6,1,2,1,7,1,4,1,3,1,1,4,9,2,
%T A120454 2,1,10,5,3,1,4,1,6,2,12,1,2,1,3,6,7,1,2,3,4,7,15,1,3,1,16,3,1,3,6,1,9,
%U A120454 8,4,1,2,1,19,2,10,2,7,1,3,1,21,1,4,4,22,10,6,1,3,2,12,11,24,4,2,1,4,4
%N A120454 Ceiling[GPF(n)/LPF(n)] where GPF is greatest prime factor, LPF is least prime factor.
%C A120454 Given GPF(n) and LPF(n), the sum is A074320, the difference is A046665, and the product is A066048. a(n) = 1 iff n is p^k iff n is in A000961.
%F A120454 a(n) = ceiling[A006530(n)/A020639(n)].
%e A120454 a(26) = ceiling[GPF(26)/LPF(26)] = ceiling[13/2] = 7.
%p A120454 A120454 := proc(n) local ifs ; if n = 1 then RETURN(1) ; else ifs := ifactors(n)[2] ; RETURN( ceil(op(1,op(-1,ifs))/op(1,op(1,ifs))) ) ; fi ; end ; for n from 1 to 100 do printf("%d, ",A120454(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 16 2006
%Y A120454 Cf. A000040, A006530, A020639, A074320, A046665, A066048.
%Y A120454 Adjacent sequences: A120451 A120452 A120453 this_sequence A120455 A120456 A120457
%Y A120454 Sequence in context: A101491 A032436 A073408 this_sequence A076511 A099244 A014671
%K A120454 easy,nonn
%O A120454 1,6
%A A120454 Jonathan Vos Post (jvospost2(AT)yahoo.com), Aug 16 2006
%E A120454 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 16 2006

%I A076511
%S A076511 0,1,1,1,1,2,1,1,1,3,1,2,1,4,7,1,1,2,1,3,3,6,1,2,1,7,1,4,1,11,1,1,13,9,
%T A076511 11,2,1,10,5,3,1,5,1,6,7,12,1,2,1,3,19,7,1,2,3,4,7,15,1,11,1,16,3,1,17,
%U A076511 23,1,9,25,23,1,2,1,19,7,10,17,9,1,3,1,21,1,5,21,22,31,6,1,11,19,12,11
%N A076511 Numerator of cototient(n)/totient(n).
%F A076511 a(n) = A051953(n)/A009195(n).
%Y A076511 Cf. denominator=A076512, A000010.
%Y A076511 Adjacent sequences: A076508 A076509 A076510 this_sequence A076512 A076513 A076514
%Y A076511 Sequence in context: A032436 A073408 A120454 this_sequence A099244 A014671 A029365
%K A076511 nonn,frac
%O A076511 1,6
%A A076511 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 15 2002

%I A099244
%S A099244 1,1,2,1,1,1,3,1,2,2,1,2,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,2,2,3,
%T A099244 2,3,3,2,2,3,3,2,3,2,2,1,2,3,3,2,3,2,2,1,3,2,2,1,2,1,1,6,1,1,1,1,1,1,1,
%U A099244 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A099244 Greatest common divisor of length of n in binary representation and its number of ones.
%C A099244 a(n) = GCD(A070939(n), A000120(n));
%C A099244 a(A000225(n)) = n and a(m) < n for m < A000225(n).
%H A099244 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%Y A099244 Cf. A099248, A099247, A099249, A099245, A099246, A007088.
%Y A099244 Adjacent sequences: A099241 A099242 A099243 this_sequence A099245 A099246 A099247
%Y A099244 Sequence in context: A073408 A120454 A076511 this_sequence A014671 A029365 A134819
%K A099244 nonn
%O A099244 1,3
%A A099244 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 08 2004

%I A014671
%S A014671 1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,0,1,0,2,1,1,1,3,1,2,
%T A014671 2,3,2,3,1,5,3,3,4,6,3,5,5,7,5,7,5,9,7,7,8,12,8,11,10,12,
%U A014671 11,13,11,17,14,14,15,20,15,19,19,22,20,22,20,27,24,25
%N A014671 Poincare series (or Molien series) for mod 2 cohomology of alternating group A_7 subset A_8 acting on polynomial ring F_2[ x_1,y_1,z_1,w_1 ].
%D A014671 A. Adem, Recent developments in the cohomology of finite groups, Notices Amer. Math. Soc., 44 (1997),806-812.
%p A014671 (1+t^18+t^20+t^21+t^24+t^25+t^27+t^45)/((1-t^8)*(1-t^12)*(1-t^14)*(1-t^15));
%Y A014671 Adjacent sequences: A014668 A014669 A014670 this_sequence A014672 A014673 A014674
%Y A014671 Sequence in context: A120454 A076511 A099244 this_sequence A029365 A134819 A095136
%K A014671 nonn
%O A014671 0,21
%A A014671 Alejandro Adem (adem(AT)math.wisc.edu)

%I A029365
%S A029365 1,0,0,0,1,0,0,1,1,1,0,1,2,1,1,1,3,1,2,2,3,3,2,3,4,4,3,
%T A029365 4,6,4,5,5,7,6,6,7,9,8,7,9,11,9,10,11,13,12,12,13,16,15,
%U A029365 14,16,19,17,18,19,22,21,21,22,26,25,24,27,30,28,29,31
%N A029365 Expansion of 1/((1-x^4)(1-x^7)(1-x^9)(1-x^12)).
%Y A029365 Adjacent sequences: A029362 A029363 A029364 this_sequence A029366 A029367 A029368
%Y A029365 Sequence in context: A076511 A099244 A014671 this_sequence A134819 A095136 A105540
%K A029365 nonn
%O A029365 0,13
%A A029365 njas

%I A134819
%S A134819 1,1,1,1,1,2,3,2,1,2,5,9,11,7,3,3,10,26,40,40,21,6,5,25,77,159,217,185,
%T A134819 85,19,9,56,222,575,1031,1230,920,356,50,18,139,654,2082,4679,7437,7982,
%U A134819 5308,1804,204,35,338,1902,7244,19983,40139,57771,56437,33860,10064,832
%N A134819 Irregular triangle read by rows: the number of hydrocarbon structures that can be drawn with a given number of carbons and units of unsaturation.
%C A134819 Computed over a period of several years, and confirmed using the Molgen program.
%D A134819 Molgen, http://molgen.de/?src=documents/publications.
%H A134819 David A. Consiglio, Jr., <a href="http://www.research.att.com/~njas/sequences/b134819.txt">Table of n, a(n) for n = 1..99</a>
%e A134819 Triangle begins:
%e A134819 1
%e A134819 1 1 1
%e A134819 1 2 3 2 1
%e A134819 2 5 9 11 7 3
%e A134819 etc.
%e A134819 a(3,1) = 2 because there are two structures that contain 3 carbons and have 1 unit of unsaturation (propene and cyclopropane)
%Y A134819 Row sums give A134818.
%Y A134819 Adjacent sequences: A134816 A134817 A134818 this_sequence A134820 A134821 A134822
%Y A134819 Sequence in context: A099244 A014671 A029365 this_sequence A095136 A105540 A057043
%K A134819 nice,nonn,tabf,new
%O A134819 1,3
%A A134819 David A. Consiglio, Jr. (davecons2002(AT)yahoo.com), Jan 28 2008, Apr 23 2008

%I A095136
%S A095136 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,3,1,2,3,2,2,1,2,2,3,1,3,2,1,2,2,4,1,3,4,
%T A095136 3,1,2,3,4,4,2,3,4,3,3,4,3,5,3,6,3,5,4,3,4,2,1,2,3,2,3,2,5,5,4,3,4,6,5,
%U A095136 2,5,3,4,1,4,2,2,3,4
%N A095136 Number of distinct prime factors of A095134(n).
%t A095136 PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1} ] & /@ FactorInteger[n]]; f[n_] := Product[ Prime[i], {i, 2, n, 2}] + Product[ Prime[i], {i, 1, n, 2}]; f[1] = 2; Table[ Length[ PrimeFactors[ f[n]]], {n, 80}]
%Y A095136 Cf. A095134.
%Y A095136 Adjacent sequences: A095133 A095134 A095135 this_sequence A095137 A095138 A095139
%Y A095136 Sequence in context: A014671 A029365 A134819 this_sequence A105540 A057043 A070091
%K A095136 nonn
%O A095136 1,12
%A A095136 Robert G. Wilson v (rgwv(AT)rgwv.com), May 27 2004

%I A105540
%S A105540 1,1,1,1,2,1,1,1,3,1,2,4,1,4,1,1,4,9,1,5,1,1,1,6,16,1,6,1,1,6,1,8,25,1,
%T A105540 7,1,3,8,21,1,10,36,1,8,1,1,6,21,52,1,12,49,1,9,1,2,1,9,40,105,1,14,64,
%U A105540 1,10,1,4,13,1,12,65,186,1,16,81,1,11,1,1,14,51,1,15,96,301,1,18,100,1
%N A105540 Triangular matrix T, read by rows, where column n equals T^(n+1) when flattened (as read by rows), for n>=0.
%C A105540 The ratio of consecutive row sums seems to converge to a value near sqrt(2).
%e A105540 Triangle T begins:
%e A105540 1;
%e A105540 1,1;
%e A105540 1,2,1;
%e A105540 1,1,3,1;
%e A105540 2,4,1,4,1;
%e A105540 1,4,9,1,5,1;
%e A105540 1,1,6,16,1,6,1;
%e A105540 1,6,1,8,25,1,7,1;
%e A105540 3,8,21,1,10,36,1,8,1;
%e A105540 1,6,21,52,1,12,49,1,9,1;
%e A105540 2,1,9,40,105,1,14,64,1,10,1;
%e A105540 4,13,1,12,65,186,1,16,81,1,11,1;
%e A105540 1,14,51,1,15,96,301,1,18,100,1,12,1; ...
%e A105540 which when flattened (read by rows) forms column 0:
%e A105540 1,1,1,1,2,1,1,1,3,1,2,4,1,4,1,1,4,9,1,5,1,...
%e A105540 The matrix square, T^2 = A105542, begins:
%e A105540 1;
%e A105540 2,1;
%e A105540 4,4,1;
%e A105540 6,8,6,1;
%e A105540 13,14,14,8,1;
%e A105540 26,47,26,22,10,1; ...
%e A105540 which when flattened (read by rows) forms column 1:
%e A105540 1,2,1,4,4,1,6,8,6,1,13,14,14,8,1,...
%e A105540 The matrix cube, T^3 = A105545, begins:
%e A105540 1;
%e A105540 3,1;
%e A105540 9,6,1;
%e A105540 21,21,9,1;
%e A105540 51,54,39,12,1;
%e A105540 142,165,111,63,15,1; ...
%e A105540 which when flattened (read by rows) forms column 2:
%e A105540 1,3,1,9,6,1,21,21,9,1,51,54,39,12,1,...
%o A105540 (PARI) {T(n,k)=local(A=matrix(n+1,n+1,r,c,if(r>=c,1)),B, TK=floor((sqrt(8*(n-k)+1)-1)/2),TC);B=A; for(r=0,n, for(c=0,r,TC=floor((sqrt(8*(r-c)+1)-1)/2); B[r+1,c+1]=(A^(c+1))[TC+1,r-c-TC*(TC+1)/2+1]);A=B); return((A^(k+1))[TK+1,n-k-TK*(TK+1)/2+1])}
%Y A105540 Cf. A105541 (row sums), A105542 (matrix square), A105545 (matrix cube).
%Y A105540 Adjacent sequences: A105537 A105538 A105539 this_sequence A105541 A105542 A105543
%Y A105540 Sequence in context: A029365 A134819 A095136 this_sequence A057043 A070091 A091981
%K A105540 nonn,tabl
%O A105540 0,5
%A A105540 Paul D. Hanna (pauldhanna(AT)juno.com), Apr 13 2005

%I A057043
%S A057043 1,1,2,1,1,1,3,1,2,10,3,9,22,25,23,38,62,1,107,33,76,166,263,176,397,
%T A057043 430,227,688,811,1481,942,518,2087,2731,3318,2563,6747,6100,12993,8171,
%U A057043 6599,15585
%N A057043 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; each k is an R(i(k),j(k)) and A057043(n)=i(L(n)), where L(n) is the n-th Lucas number.
%Y A057043 Adjacent sequences: A057040 A057041 A057042 this_sequence A057044 A057045 A057046
%Y A057043 Sequence in context: A134819 A095136 A105540 this_sequence A070091 A091981 A060247
%K A057043 nonn
%O A057043 1,3
%A A057043 Clark Kimberling (ck6(AT)evansville.edu), Jul 30 2000

%I A070091
%S A070091 0,0,1,0,1,0,2,1,1,1,3,1,3,1,2,2,4,2,5,2,2,2,6,2,5,3,5,3,7,2,8,4,4,4,6,
%T A070091 3,9,4,6,4,10,4,11,5,6,5,12,4,10,5,8,6,13,4,10,6,8,7,15,4,15,7,10,8,12,
%U A070091 6,17,8,10,6,18,6,18,9,10,9,14,6,20,8,13
%N A070091 Number of isosceles integer triangles with perimeter n and relatively prime side lengths.
%C A070091 a(n) = A051493(n) - A005044(n-6).
%H A070091 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a070080.txt">Integer-sided triangles</a>
%e A070091 For n=15 there are A005044(15)=7 integer triangles: [1,7,7], [2,6,7], [3,5,7], [3,6,6], [4,4,7], [4,5,6], and [5,5,5]: four are isosceles: [1<7=7], [3<6=6], [4=4<7], and [5=5=5], but GCD(3,6,6)>1 and GCD(5,5,5)>1, therefore a(15)=2.
%Y A070091 Cf. A070080, A070081, A070082, A059169, A070099, A070107, A070084, A070116.
%Y A070091 Adjacent sequences: A070088 A070089 A070090 this_sequence A070092 A070093 A070094
%Y A070091 Sequence in context: A095136 A105540 A057043 this_sequence A091981 A060247 A060246
%K A070091 nonn
%O A070091 1,7
%A A070091 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002

%I A091981
%S A091981 1,0,0,0,0,0,1,0,2,1,1,1,3,1,3,3,4,3,5,3,7,6,7,8,12,8,12,12,14,14,18,15,
%T A091981 22,20,22,24,30,26,33,33,37,37,43,40,50,49,52,55,64,59,69,70,76,78,86,83,
%U A091981 97,96,101,106,118,113,126,128,137,140,151,149,167,167,174,181,197,192
%N A091981 Poincare series (or Molien series) for GL_2(4):2.
%D A091981 A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 269.
%F A091981 G.f. = p(x)/q(x), where
%F A091981 p(x) = 1 + x^6 + 2*x^8 + x^9 + x^11 + 2*x^12 + x^13 + 3*x^14 + 2*x^15 + 3*x^16 + 3*x^17 + 2*x^18 + 2*x^19 + 4*x^20 + 3*x^21 + 4*x^22 + 4*x^23 + 4*x^24 + 4*x^25 + 4*x^26 + 4*x^27 + 5*x^28 + 5*x^29 + 4*x^30 + 4*x^31 + 4*x^32 + 4*x^33 + 4*x^34 + 4*x^35 + 3*x^36 + 4*x^37 + 2*x^38 + 2*x^39 + 3*x^40 + 3*x^41 + 2*x^42 + 3*x^43 + x^44 + 2*x^45 + x^46 + x^48 + 2*x^49 + x^51 + x^57,
%F A091981 and q(x) = (1-x^10)*(1-x^12)*(1-x^15)*(1-x^24).
%Y A091981 Adjacent sequences: A091978 A091979 A091980 this_sequence A091982 A091983 A091984
%Y A091981 Sequence in context: A105540 A057043 A070091 this_sequence A060247 A060246 A123541
%K A091981 nonn
%O A091981 0,9
%A A091981 njas, Mar 18 2004

%I A060247
%S A060247 1,1,2,1,1,1,3,1,3,3,4,5,1,3,3,4,5,1,3,3,6,7,8,1,7,7,7,7,8,9,9,9,1,
%T A060247 5,5,8,8,9,10,1,5,5,10,10,11,12,12
%N A060247 Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,q) as q runs through the primes and prime powers.
%D A060247 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.
%e A060247 1,1,2; 1,1,1,3; 1,3,3,4,5; 1,3,3,4,5; ... (for q = 2,3,4,5, ...).
%o A060247 (MAGMA) CharacterTable(PSL(2,7)); (say)
%Y A060247 Cf. A060246, A060240, A060241.
%Y A060247 Adjacent sequences: A060244 A060245 A060246 this_sequence A060248 A060249 A060250
%Y A060247 Sequence in context: A057043 A070091 A091981 this_sequence A060246 A123541 A090379
%K A060247 tabf,nonn,nice,easy,more
%O A060247 1,3
%A A060247 njas, Mar 22 2001

%I A060246
%S A060246 1,1,2,1,1,1,3,1,3,3,4,5,1,3,3,6,7,8,1,5,5,10,10,11,12,12
%N A060246 Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,q) as q runs through the primes.
%D A060246 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.
%e A060246 1,1,2; 1,1,1,3; 1,3,3,4,5; ... (for q=2,3,5,...).
%o A060246 (MAGMA) CharacterTable(PSL(2,7)); (say)
%Y A060246 Cf. A060247, A060240, A060241.
%Y A060246 Adjacent sequences: A060243 A060244 A060245 this_sequence A060247 A060248 A060249
%Y A060246 Sequence in context: A070091 A091981 A060247 this_sequence A123541 A090379 A077254
%K A060246 tabf,nonn,nice,easy,more
%O A060246 1,3
%A A060246 njas, Mar 22 2001

%I A123541
%S A123541 2,1,1,1,3,1,4,1,1,2,7,2,7,1,1,4,11,3,9,2,4,4,11,0,2,4,4,11,11,6,14,2,5,
%T A123541 7,6,8,16,10,4,15,13,9,13,10,5,9,14,5,9,9,11,10,17,6,9,11,13,19,20,11,22,
%U A123541 8,17,14,13,14,20,13,13,22,23,9,20,8,12,16,11,13,21,13,13,16,14,12,16,15
%N A123541 a(0) = 2; for n > 0, a(n) = number of earlier terms which when added to n give a prime.
%H A123541 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b123541.txt">Table of n, a(n) for n=0..2000</a>
%F A123541 a(n) = A114897(n+1) for n>2. - T. D. Noe (noe(AT)sspectra.com), Apr 30 2007
%p A123541 M:=100; a:=array(0..M); a[0]:=2; for n from 1 to M do t1:=0; for i from 0 to n-1 do if isprime(n+a[i]) then t1:=t1+1; fi; od: a[n]:=t1; od: [seq(a[n],n=0..M)];
%t A123541 t={2}; Do[AppendTo[t,Length[Select[t+n,PrimeQ]]], {n,2000}]; t - T. D. Noe (noe(AT)sspectra.com), Apr 30 2007
%Y A123541 Cf. A114899
%Y A123541 Adjacent sequences: A123538 A123539 A123540 this_sequence A123542 A123543 A123544
%Y A123541 Sequence in context: A091981 A060247 A060246 this_sequence A090379 A077254 A074761
%K A123541 nonn
%O A123541 0,1
%A A123541 njas, based on email from Zak Seidov, Oct 16 2006

%I A090379
%S A090379 1,1,0,0,1,0,2,1,1,1,3,1,4,2,2,2,5,2,6,3,3,3,7,3,3,3,3,3,8,4,9,4,4,4,4,
%T A090379 4,10,5,5,5,11,5,12,6,6,6,13,6,6,6,6,6,14,7,7,7,7,7,15,7,16,8,8,8,8,8,
%U A090379 17,8,8,8,18,9,19,9,9,9,9,9,20,10,10,10,21,10,10,10,10,10,22,11,11,11
%N A090379 a(n) = Abs[Floor[(PrimePi[n]-2)/(2-PrimePi[n]+PrimePi[n-1])]]
%C A090379 A chaotic function based on the distribution of primes and the information in the composite numbers.
%t A090379 f[n_] := Abs[Floor[(PrimePi[n] - 2)/(2 - PrimePi[n] + PrimePi[n - 1])]]; Table[ f[n], {n, 1, 92}]
%Y A090379 Adjacent sequences: A090376 A090377 A090378 this_sequence A090380 A090381 A090382
%Y A090379 Sequence in context: A060247 A060246 A123541 this_sequence A077254 A074761 A037861
%K A090379 nonn
%O A090379 1,7
%A A090379 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 29 2004

%I A077254
%S A077254 0,1,2,1,1,1,3,1,8,1,9,1,2,1,8,1,8,1,10,1,13,15,14,1,7,9,1,9,22,19,3,1,
%T A077254 26,9,4,1,9,7,5,1,15,1,19,9,17,41,23,1,31,1,11,1,29,1,23,9,8,13,41,1,
%U A077254 39,41,55,1,53,31,63,13,8,1,69,1,2,9,49,5,16,25,6,1,80,39,16,1,29,83
%N A077254 Prime(n)^n mod n.
%C A077254 a(A077255(n)) = 1.
%e A077254 a(13) = prime(13)^13 mod 13 = 41^13 mod 13 = 925103102315013629321 mod 13 = 2.
%Y A077254 a(n) = A062457(n) mod n, A077256, A000040, A000027.
%Y A077254 Adjacent sequences: A077251 A077252 A077253 this_sequence A077255 A077256 A077257
%Y A077254 Sequence in context: A060246 A123541 A090379 this_sequence A074761 A037861 A072410
%K A077254 nonn
%O A077254 1,3
%A A077254 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 31 2002

%I A074761
%S A074761 1,1,1,1,1,2,1,1,1,3,1,9,1,4,5,1,1,12,1,27,7,6,1,81,1,7,1,54,1,407,1,1,
%T A074761 11,9,13,494,1,10,13,423,1,981,1,137,115,12,1,1309,1,59,17,193,1,240,
%U A074761 21,1207,19,15,1,47274,1,16,239,1,25,3284,1,333,23,3731,1,42109,1,19
%N A074761 Number of partitions of n of order n.
%C A074761 Order of partition is lcm of its parts.
%F A074761 Coefficient of x^n in expansion of Sum_{i divides n} mu(n/i)*1/Product_{j divides i} (1-x^j).
%Y A074761 Cf. A018818, A074351, A074752.
%Y A074761 Adjacent sequences: A074758 A074759 A074760 this_sequence A074762 A074763 A074764
%Y A074761 Sequence in context: A123541 A090379 A077254 this_sequence A037861 A072410 A052409
%K A074761 easy,nonn
%O A074761 1,6
%A A074761 Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 28 2002

%I A037861
%S A037861 1,1,0,2,1,1,1,3,2,0,0,2,0,2,2,4,3,1,1,1,1,1,1,3,1,
%T A037861 1,1,3,1,3,3,5,4,2,2,0,2,0,0,2,2,0,0,2,0,2,2,4,2,
%U A037861 0,0,2,0,2,2,4,0,2,2,4,2,4,4,6,5,3,3,1,3,1,1,1,3
%V A037861 1,-1,0,-2,1,-1,-1,-3,2,0,0,-2,0,-2,-2,-4,3,1,1,-1,1,-1,-1,-3,1,
%W A037861 -1,-1,-3,-1,-3,-3,-5,4,2,2,0,2,0,0,-2,2,0,0,-2,0,-2,-2,-4,2,
%X A037861 0,0,-2,0,-2,-2,-4,0,-2,-2,-4,-2,-4,-4,-6,5,3,3,1,3,1,1,-1,3
%N A037861 (Number of 0's)-(number of 1's) in base 2 representation of n.
%C A037861 Sum(n=1 to infinity, a(n)/((2n)(2n+1))) = log Pi/4 = -0.24156... . - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 01 2005
%D A037861 J. Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analogue of Hadjicostas's formula, Amer. Math. Monthly 112 (2005) 61-65.
%H A037861 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/b037861.txt">Table of n, a(n) for n = 0..10000</a>
%H A037861 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A037861 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Table of generating functions</a>
%H A037861 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>
%H A037861 J. Sondow, <a href="http://arXiv.org/abs/math.NT/0508042">New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)</a>
%F A037861 a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n); a(2n) = a(n)+1; a(2n+1) = a(2n) - 2 = a(n) - 1: Henry Bottomley (se16(AT)btinternet.com), Oct 27 2000
%F A037861 G.f. satisfies A(x) = (1+x)A(x^2) - x(2+x)/(1+x) - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 26 2006
%F A037861 a(n) = b(n) for n>0 with b(0)=0 and b(n) = b(floor(n/2)) + (-1)^(n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 31 2007
%t A037861 Table[ Abs[ Count[ IntegerDigits[n, 2], 0] - Count[ IntegerDigits[n, 2], 1] ], {n, 0, 75} ]
%Y A037861 Cf. A031443 for n when a(n)=0, A053738 for n when a(n) odd, A053754 for n when a(n) even, A030300 for a(n+1) mod 2.
%Y A037861 Cf. A094640, A110625.
%Y A037861 Adjacent sequences: A037858 A037859 A037860 this_sequence A037862 A037863 A037864
%Y A037861 Sequence in context: A090379 A077254 A074761 this_sequence A072410 A052409 A051904
%K A037861 base,sign
%O A037861 0,4
%A A037861 Clark Kimberling (ck6(AT)evansville.edu)

%I A072410
%S A072410 0,1,1,2,1,1,1,3,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,3,1,1,1,1,5,1,1,1
%N A072410 Duplicate of A052409.
%Y A072410 Adjacent sequences: A072407 A072408 A072409 this_sequence A072411 A072412 A072413
%Y A072410 Sequence in context: A077254 A074761 A037861 this_sequence A052409 A051904 A070012
%K A072410 dead
%O A072410 1,4

%I A052409
%S A052409 0,1,1,2,1,1,1,3,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,3,1,1,1,1,5,1,1,1,
%T A052409 2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1,
%U A052409 1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1
%N A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).
%C A052409 Greatest common divisor of all prime-exponents in canonical factorization of n for n>1: a(n)>1 iff n is a perfect power; a(A001597(k))=A025479(k). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 13 2002
%H A052409 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Power.html">Power</a>
%H A052409 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>
%e A052409 n=72=2.2.2.3.3: GCD[exponents]=GCD[3,2]=1. It deviates from Min of exponents(A051904).
%t A052409 Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (*Chandler*)
%Y A052409 Cf. A052410, A005361, A051903, A051904, A072411-A072414.
%Y A052409 Adjacent sequences: A052406 A052407 A052408 this_sequence A052410 A052411 A052412
%Y A052409 Sequence in context: A074761 A037861 A072410 this_sequence A051904 A070012 A071178
%K A052409 nonn
%O A052409 1,4
%A A052409 Eric Weisstein (eric(AT)weisstein.com)
%E A052409 More terms from Labos E. (labos(AT)ana.sote.hu), Jun 17 2002

%I A051904
%S A051904 0,1,1,2,1,1,1,3,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,3,1,1,1,1,5,1,1,1,
%T A051904 2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1,
%U A051904 1,2,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1
%N A051904 Minimal exponent in prime factorization of n.
%H A051904 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NivensConstant.html">Niven's Constant</a>
%H A051904 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NivensConstant.html">Niven's Constant</a>
%e A051904 For n = 72 = 2^3*3^2, a(72) = Min(exponents) = Min(3,2) = 2.
%t A051904 Table[If[n == 1, 0, Min @@ Last /@ FactorInteger[n]], {n, 100}] (*Chandler*)
%Y A051904 Cf. A005361, A008479, A051903, A052409.
%Y A051904 Adjacent sequences: A051901 A051902 A051903 this_sequence A051905 A051906 A051907
%Y A051904 Sequence in context: A037861 A072410 A052409 this_sequence A070012 A071178 A072776
%K A051904 nonn,easy
%O A051904 1,4
%A A051904 Labos E. (labos(AT)ana.sote.hu), Dec 16 1999

%I A070012
%S A070012 1,1,2,1,1,1,3,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2,2,1,3,1,1,1,1,5,1,1,1,2,
%T A070012 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,2,1,2,1,1,1,1,1,1,1,6,1,1,1,1,1,1,1,
%U A070012 2,1,1,1,1,1,1,1,2,4,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,2,1,1,1,2,1,1
%N A070012 Floor of number of prime factors of n divided by the number of n's distinct prime factors.
%C A070012 a(n) is the integer part of the average of the exponents in the prime factorization of n.
%F A070012 a(n) = floor(bigomega(n)/omega(n)) for n>=2.
%e A070012 a(12)=1 because 12=2^2 * 3^1 and floor(bigomega(12)/omega(12))=floor((2+1)/2)=1. a(36)=2 because 36=2^2 * 3^2 and floor(bigomega(36)/omega(36))=floor((2+2)/2)=2. a(60)=1 because 60=2^2 * 3^1 * 5^1 and floor(bigomega(60)/omega(60))= floor((2+1+1)/3)=1. 36 is in A067340. 12 and 60 are in A070011.
%o A070012 (PARI) v=[]; for(n=2,150,v=concat(v,(bigomega(n)\omega(n)))); v In PARI, j\k using the "\" operator for integers j,k is equivalent to floor(j/k).
%Y A070012 Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is integer before floor applied), A070011 (ratio is not an integer), A070013 (ratio rounded), A070014 (ceiling of ratio), A046660 (bigomega(n)-omega(n)).
%Y A070012 Adjacent sequences: A070009 A070010 A070011 this_sequence A070013 A070014 A070015
%Y A070012 Sequence in context: A072410 A052409 A051904 this_sequence A071178 A072776 A077481
%K A070012 nonn
%O A070012 2,3
%A A070012 Rick L. Shepherd (rshepherd2(AT)hotmail.com), Apr 11 2002

%I A071178
%S A071178 0,1,1,2,1,1,1,3,2,1,1,1,1,1,1,4,1,2,1,1,1,1,1,1,2,1,3,1,1,1,1,5,1,1,1,
%T A071178 2,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,3,1,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1,
%U A071178 1,2,1,1,2,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1
%N A071178 Exponent of the largest prime factor of n.
%Y A071178 Cf. A067029.
%Y A071178 Adjacent sequences: A071175 A071176 A071177 this_sequence A071179 A071180 A071181
%Y A071178 Sequence in context: A052409 A051904 A070012 this_sequence A072776 A077481 A086290
%K A071178 easy,nonn
%O A071178 1,4
%A A071178 Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 10 2002

%I A072776
%S A072776 1,1,1,2,1,1,1,3,2,1,1,1,1,1,4,1,1,1,1,1,2,1,3,1,1,1,5,1,1,1,2,1,1,1,1,
%T A072776 1,1,1,1,2,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,
%U A072776 1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,3,1,7,1,1,1,1,1,1,1,1,1,1,1,1
%N A072776 Exponents of powers of square-free numbers.
%C A072776 A072774(n) = A072775(n)^a(n);
%C A072776 A072774(n) is square-free iff a(n)=1.
%Y A072776 Adjacent sequences: A072773 A072774 A072775 this_sequence A072777 A072778 A072779
%Y A072776 Sequence in context: A051904 A070012 A071178 this_sequence A077481 A086290 A136568
%K A072776 nonn
%O A072776 1,4
%A A072776 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 10 2002

%I A077481
%S A077481 1,1,2,1,1,1,3,2,1,1,1,1,1,4,1,1,1,1,1,2,2,1,3,1,1,1,5,1,1,1,2,1,1,1,2,
%T A077481 1,1,1,1,1,2,1,1,2,1,2,1,1,1,1,1,6,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,2,
%U A077481 1,1,1,1,1,3,1,2,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,3,1,7,1,1,1,1,1,2
%N A077481 Quotient A077480(n) / A077479(n).
%Y A077481 Cf. A067340, A001221, A001222.
%Y A077481 Adjacent sequences: A077478 A077479 A077480 this_sequence A077482 A077483 A077484
%Y A077481 Sequence in context: A070012 A071178 A072776 this_sequence A086290 A136568 A039961
%K A077481 nonn
%O A077481 1,3
%A A077481 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 06 2002

%I A086290
%S A086290 0,1,1,2,1,1,1,3,2,1,1,1,1,4,1,1,1,1,2,3,1,1,5,1,2,1,1,1,1,2,1,1,1,1,1,
%T A086290 6,1,2,1,1,4,1,1,1,1,2,1,2,1,1,3,1,7,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,3,1,
%U A086290 2,1,5,1,1,1,8,1,1,2,1,1,1,1,2,1,3,1,1,1,1,1,2,2,1,1,3,2,1,1
%N A086290 Minimal exponent in prime factorization of 7-smooth numbers.
%C A086290 a(n) = A051904(A002473(n));
%C A086290 a(n) <= A086291(n) <= A086289(n).
%Y A086290 Adjacent sequences: A086287 A086288 A086289 this_sequence A086291 A086292 A086293
%Y A086290 Sequence in context: A071178 A072776 A077481 this_sequence A136568 A039961 A108299
%K A086290 nonn
%O A086290 1,4
%A A086290 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 15 2003

%I A136568
%S A136568 0,1,1,2,1,1,1,3,2,1,1,1,2,1,1,1,4,1,1,2,1,1,2,1,1,1,1,3,2,1,3,1,2,1,1,
%T A136568 1,5,1,1,1,2,1,1,1,1,3,1,1,1,1,2,1,2,1,1,1,4,2,1,2
%N A136568 Irregular triangle read by rows: row n contains the distinct values (in numerical order) making up the non-zero exponents in the prime-factorization of n. (Row 1 = (0).).
%C A136568 Row n, for n>=2, contains A071625(n) terms.
%e A136568 4200 = 2^3 * 3^1 * 5^2 * 7^1. The exponents of the prime factorization are therefore 3,1,2,1. Therefore row 4200 is (1,2,3).
%Y A136568 Cf. A071625, A136565.
%Y A136568 Adjacent sequences: A136565 A136566 A136567 this_sequence A136569 A136570 A136571
%Y A136568 Sequence in context: A072776 A077481 A086290 this_sequence A039961 A108299 A065941
%K A136568 more,nonn,tabf
%O A136568 1,4
%A A136568 Leroy Quet (qq-quet(AT)mindspring.com), Jan 07 2008

%I A039961
%S A039961 1,1,1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,3,3,1
%V A039961 1,1,1,-1,1,-1,-1,1,-1,-2,1,1,-1,-3,2,1,1,-1,-4,3,3,-1
%N A039961 Triangle of coefficients in a Fibonacci-like sequence of polynomials.
%D A039961 A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.
%F A039961 q_{n+2}(x)=x*q_{n+1)(x)-q_n(x), q_1(x)=q_2(x)=1.
%e A039961 1; 1; 1 -1; 1 -1 -1; 1 -1 -2 1; 1 -1 -3 2 1; ...
%Y A039961 Adjacent sequences: A039958 A039959 A039960 this_sequence A039962 A039963 A039964
%Y A039961 Sequence in context: A077481 A086290 A136568 this_sequence A108299 A065941 A123320
%K A039961 sign
%O A039961 1,10
%A A039961 njas

%I A108299
%S A108299 1,1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,3,3,1,1,1,5,4,6,3,1,1,1,6,5,10,6,4,
%T A108299 1,1,1,7,6,15,10,10,4,1,1,1,8,7,21,15,20,10,5,1,1,1,9,8,28,21,35,20,15,
%U A108299 5,1,1,1,10,9,36,28,56,35,35,15,6,1,1,1,11,10,45,36,84,56,70
%V A108299 1,1,-1,1,-1,-1,1,-1,-2,1,1,-1,-3,2,1,1,-1,-4,3,3,-1,1,-1,-5,4,6,-3,-1,1,-1,-6,5,10,-6,
%W A108299 -4,1,1,-1,-7,6,15,-10,-10,4,1,1,-1,-8,7,21,-15,-20,10,5,-1,1,-1,-9,8,28,-21,-35,20,15,
%X A108299 -5,-1,1,-1,-10,9,36,-28,-56,35,35,-15,-6,1,1,-1,-11,10,45,-36,-84,56,70
%N A108299 Triangle read by rows, 0 <= k <= n: T(n,k)=binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
%C A108299 Let L(n,x) = Sum(T(n,k)*x^(n-k): 0<=k<=n) and Pi=3.14...:
%C A108299 L(n,x) = Prod(x - 2*cos((2*k-1)*Pi/(2*n+1)): 1<=k<=n);
%C A108299 Sum(T(n,k): 0<=k<=n) = L(n,1) = A010892(n+1);
%C A108299 Sum(abs(T(n,k)): 0<=k<=n) = A000045(n+2);
%C A108299 abs(T(n,k))=A065941(n,k), T(n,k)=A065941(n,k)*A087960(k);
%C A108299 T(2*n,k) + T(2*n+1,k+1) = 0 for 0<=k<=2*n;
%C A108299 T(n,0)=A000012(n)=1; T(n,1)=-1 for n>0;
%C A108299 T(n,2)=-(n-1) for n>1; T(n,3)=A000027(n)=n for n>2;
%C A108299 T(n,4)=A000217(n-3) for n>3; T(n,5)=-A000217(n-4) for n>4;
%C A108299 T(n,6)=-A000292(n-5) for n>5; T(n,7)=A000292(n-6) for n>6;
%C A108299 T(n,n-3)=A058187(n-3)*(-1)^[n/2] for n>2;
%C A108299 T(n,n-2)=A008805(n-2)*(-1)^[(n+1)/2] for n>1;
%C A108299 T(n,n-1)=A008619(n-1)*(-1)^[n/2] for n>0;
%C A108299 T(n,n) = L(n,0) = (-1)^[(n+1)/2];
%C A108299 L(n,1) = A010892(n+1); L(n,-1) = A061347(n+2);
%C A108299 L(n,2) = 1; L(n,-2) = A005408(n)*(-1)^n;
%C A108299 L(n,3) = A001519(n); L(n,-3) = A002878(n)*(-1)^n;
%C A108299 L(n,4) = A001835(n+1); L(n,-4) = A001834(n)*(-1)^n;
%C A108299 L(n,5) = A004253(n); L(n,-5) = A030221(n)*(-1)^n;
%C A108299 L(n,6) = A001653(n); L(n,-6) = A002315(n)*(-1)^n;
%C A108299 L(n,7) = A049685(n); L(n,-7) = A033890(n)*(-1)^n;
%C A108299 L(n,8) = A070997(n); L(n,-8) = A057080(n)*(-1)^n;
%C A108299 L(n,9) = A070998(n); L(n,-9) = A057081(n)*(-1)^n;
%C A108299 L(n,10) = A072256(n+1); L(n,-10) = A054320(n)*(-1)^n;
%C A108299 L(n,11) = A078922(n+1); L(n,-11) = A097783(n)*(-1)^n;
%C A108299 L(n,12) = A077417(n); L(n,-12) = A077416(n)*(-1)^n;
%C A108299 L(n,13) = A085260(n);
%C A108299 L(n,14) = A001570(n); L(n,-14) = A028230(n)*(-1)^n;
%C A108299 L(n,n) = A108366(n); L(n,-n) = A108367(n).
%C A108299 Row n of the matrix inverse has g.f.: x^[n/2]*(1-x)^(n-[n/2]). - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 12 2005
%D A108299 Friedrich L. Bauer, 'De Moivre und Lagrange: Cosinus eines rationalen Vielfachen von Pi', Informatik Spektrum 28 (Springer, 2005).
%F A108299 T(n+1, k) = if sign(T(n, k-1))=sign(T(n, k)) then T(n, k-1)+T(n, k) else -T(n, k-1) for 0<k<n, T(n, 0) = 1, T(n, n) = (-1)^[(n+1)/2].
%F A108299 G.f.: A(x, y) = (1 - x*y)/(1 - x + x^2*y^2). - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 12 2005
%e A108299 Matrix inverse begins:
%e A108299 1;
%e A108299 1,-1;
%e A108299 0,1,-1;
%e A108299 0,1,-2,1;
%e A108299 0,0,1,-2,1;
%e A108299 0,0,1,-3,3,-1;
%e A108299 0,0,0,1,-3,3,-1; ...
%o A108299 (PARI) {T(n,k)=polcoeff(polcoeff((1-x*y)/(1-x+x^2*y^2+x^2*O(x^n)),n,x)+y*O(y^k),k,y)} (Hanna)
%Y A108299 Cf. A049310, A039961.
%Y A108299 Adjacent sequences: A108296 A108297 A108298 this_sequence A108300 A108301 A108302
%Y A108299 Sequence in context: A086290 A136568 A039961 this_sequence A065941 A123320 A054123
%K A108299 sign,tabl
%O A108299 0,9
%A A108299 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

%I A065941
%S A065941 1,1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,3,3,1,1,1,5,4,6,3,1,1,1,6,5,10,6,
%T A065941 4,1,1,1,7,6,15,10,10,4,1,1,1,8,7,21,15,20,10,5,1,1,1,9,8,28,21,35,20,
%U A065941 15,5,1,1,1,10,9,36,28,56,35,35,15,6,1,1,1,11,10,45,36,84,56,70,35,21
%N A065941 Triangle T[n,k] := binomial(n-floor((k+1)/2),floor(k/2))
%C A065941 Row sums give the Fibonacci sequence. So do the alternating row sums.
%C A065941 Triangle of coefficients of polynomials defined by p(0,x)=p(1,x)=1, p(n+2,x)=x*p(n+1,x)+p(n,x) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 08 2005
%C A065941 T(n,k) = A108299(n,k)*A087960(k) = abs(A108299(n,k)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005
%D A065941 Henry W. Gould, "A Variant of Pascal's Triangle", The Fibonacci Quarterly,3;4 Dec. 1965, pp. 257-271.
%D A065941 Jay Kappraff, "Beyond Measure, A Guided Tour Through Nature, Myth, and Number", World Scientific, 2002; p. 490.
%D A065941 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)
%D A065941 Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
%H A065941 E. Munarini and N. Z. Salvi, <a href="http://www.mat.univie.ac.at/~slc/opapers/s49zagaglia.html">Binary strings without zigzags</a>
%F A065941 As a square array read by antidiagonals, this is given by T1(n, k) = binomial(floor(n/2) + k, k) - Paul Barry (pbarry(AT)wit.ie), Mar 11 2003
%F A065941 Triangle is a reflection of that in A066170 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 16 2004
%F A065941 Recurrences: T(k, 0) = 1, T(k, n) = T(k-1, n) + T(k-2, n-2), or T(k, n) = T(k-1, n) + T(k-1, n-1) if n even, T(k-1, n-1) if n odd. - Ralf Stephan, May 17 2004
%F A065941 G.f.: sum[n, sum[k, T(k, n)x^ky^n]] = (1+xy)/(1-y-x^2y^2). sum[n>=0, T(k, n)y^n] = y^k/(1-y)^[k/2]. - Ralf Stephan, May 17 2004
%e A065941 1; 1,1; 1,1,1; 1,1,2,1; 1,1,3,2,1; ...
%Y A065941 A065942 (central stalk sequence), A000045 (row sums)
%Y A065941 Cf. A066170, A006356, A006357, A084534.
%Y A065941 Reflected version of A046854.
%Y A065941 Adjacent sequences: A065938 A065939 A065940 this_sequence A065942 A065943 A065944
%Y A065941 Sequence in context: A136568 A039961 A108299 this_sequence A123320 A054123 A119269
%K A065941 nonn,tabl
%O A065941 0,9
%A A065941 Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 29 2001

%I A123320
%S A123320 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,4,2,1,1,1,5,6,4,2,1,1,1,6,6,3,4,2,1,
%T A123320 1,1,7,12,10,3,4,2,1,1,1,8,8,6,6,3,4,2,1,1,1,9,20,20,9,6,3,4,2,1,1,1,10,
%U A123320 10,21,8,10,6,3,4,2,1,1,1,11,30,24,20,11,10,6,3,4,2,1,1,1,12,12,35,9,12
%N A123320 Table of the cycle lengths for "imperfect" (generalized) faro shuffles with cut of size k returning a deck of size n to its original order.
%C A123320 An "imperfect" (generalized) faro shuffle with cut of size k for a deck of size n is performed by first cutting the deck into a top pile of k cards and a bottom pile of n-k cards, performing a perfect faro shuffle on the bottomost min(k,n-k) cards of each pile, and placing any remaining cards on top of the deck. (Thus k may range from 0 to n inclusive, hence the offset is 0). The central column T(2k,k) gives the "perfect" faro shuffle cycles A002326.
%H A123320 Marc LeBrun, <a href="http://www.research.att.com/~njas/sequences/b123320.txt">First 100 rows, flattened</a>
%e A123320 T(5,2)=4 because the (5,2) shuffle cycles with period 4:
%e A123320 12345 --> 31425 --> 43215 --> 24135 --> 12345 etc.
%Y A123320 Cf. A002326.
%Y A123320 Adjacent sequences: A123317 A123318 A123319 this_sequence A123321 A123322 A123323
%Y A123320 Sequence in context: A039961 A108299 A065941 this_sequence A054123 A119269 A129713
%K A123320 easy,nonn,tabl
%O A123320 0,8
%A A123320 Marc LeBrun (mlb(AT)well.com), Sep 25 2006

%I A054123
%S A054123 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,4,2,1,1,1,5,7,4,2,1,1,1,6,11,8,4,2,1,
%T A054123 1,1,7,16,15,8,4,2,1,1,1,8,22,26,16,8,4,2,1,1,1,9,29,42,31,16,8,4,2,1,1,
%U A054123 1,10,37,64,57,32,16,8,4,2,1,1,1,11,46,93,99,63,32,16,8,4,2,1,1,1,12,56
%N A054123 Right Fibonacci row-sum array T(n,k), n >= 0, 0<=k<=n.
%H A054123 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pas.html#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A054123 T(n, 0)=T(n, n)=1 for n >= 0; T(n, k)=T(n-1, k)+T(n-2, k-1) for k=1, 2, ..., n-1, n >= 2.
%F A054123 T(n, k)=T(n-1, k-1)+U(n-1, k) for k=1, 2, ..., [n/2], n >= 3, array U as in A011973.
%e A054123 Rows: {1}, {1,1}, {1,1,1}, {1,2,1,1}, {1,3,2,2,1}, ...
%Y A054123 Reflection of array in A054124 about vertical central line.
%Y A054123 Row sums: 1, 2, 3, 5, 8, 13, ... (Fibonacci numbers, A000045). Central numbers: 1, 1, 2, 4, 8, ... (binary powers, A000079). Cf. A011973.
%Y A054123 Adjacent sequences: A054120 A054121 A054122 this_sequence A054124 A054125 A054126
%Y A054123 Sequence in context: A108299 A065941 A123320 this_sequence A119269 A129713 A096669
%K A054123 nonn,tabl,eigen,easy,nice
%O A054123 0,8
%A A054123 Clark Kimberling (ck6(AT)evansville.edu)
%E A054123 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003

%I A119269
%S A119269 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,4,2,1,1,1,6,6,4,2,1,1,1,8,11,7,4,2,1,
%T A119269 1,1,12,19,13,7,4,2,1,1,1,16,33,25,14,7,4,2,1,1,1,22,55,49,27,14,7,4,2,
%U A119269 1,1,1,29,95,93,55,28,14,7,4,2,1,1,1,40,158,181,111,57,28,14,7,4,2,1,1
%N A119269 Table by anti-diagonals: number of m-dimensional partitions of n up to conjugacy, for n >= 1, m >= 0.
%C A119269 Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.
%C A119269 Transposed table is A119338. - Max Alekseyev (maxal(AT)cs.ucsd.edu), May 14 2006
%F A119269 a(n,m) = a(n,n-2) for m >= n-1.
%e A119269 Table starts:
%e A119269 1,1, 1, 1, 1
%e A119269 1,1, 1, 1, 1
%e A119269 1,2, 2, 2, 2
%e A119269 1,3, 4, 4, 4
%e A119269 1,4, 6, 7, 7
%e A119269 1,6,11,13,14
%Y A119269 Columns A005987, A000786, A119266, A119267; diagonal A119268. Cf. A096751, A119270.
%Y A119269 Cf. A119339, A119340, A119341, A119342.
%Y A119269 Adjacent sequences: A119266 A119267 A119268 this_sequence A119270 A119271 A119272
%Y A119269 Sequence in context: A065941 A123320 A054123 this_sequence A129713 A096669 A096591
%K A119269 nonn,tabl
%O A119269 1,8
%A A119269 Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 11 2006
%E A119269 More terms from Max Alekseyev (maxal(AT)cs.ucsd.edu), May 14 2006

%I A129713
%S A129713 1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,5,3,2,1,1,1,8,5,3,2,1,1,1,13,8,5,3,2,1,1,
%T A129713 1,21,13,8,5,3,2,1,1,1,34,21,13,8,5,3,2,1,1,1,55,34,21,13,8,5,3,2,1,1,1,
%U A129713 89,55,34,21,13,8,5,3,2,1,1,1,144,89,55,34,21,13,8,5,3,2,1,1,1,233,144
%N A129713 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 1's (0<=k<=n). A Fibonacci binary word is a binary word having no 00 subword.
%C A129713 Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), 0<=k<=n)=F(n+3)-2=A001911(n).
%F A129713 T(n,k)=F(n-k) if k<=n-2, T(n,n-1)=T(n,n)=1, where F(j) are the Fibonacci numbers (F(0)=0, F(1)=1). G.f.=G(t,z)=(1-z^2)/[(1-z-z^2)(1-tz)].
%e A129713 T(6,2)=3 because we have 110110, 110111, 110101.
%e A129713 Triangle starts:
%e A129713 1;
%e A129713 1,2;
%e A129713 1,1,1;
%e A129713 2,1,1,1;
%e A129713 3,2,1,1,1;
%e A129713 5,3,2,1,1,1;
%e A129713 8,5,3,2,1,1,1;
%p A129713 with(combinat): T:=proc(n,k) if k<=n-2 then fibonacci(n-k) elif k=n-1 or k=n then 1 else 0 fi end: for n from 0 to 15 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A129713 Cf. A000045, A001911.
%Y A129713 Adjacent sequences: A129710 A129711 A129712 this_sequence A129714 A129715 A129716
%Y A129713 Sequence in context: A123320 A054123 A119269 this_sequence A096669 A096591 A115568
%K A129713 nonn,tabl
%O A129713 0,7
%A A129713 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 12 2007

%I A096669
%S A096669 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,5,5,2,1,1,1,8,9,5,2,1,1,1,13,18,12,5,2,
%T A096669 1,1,1,21,37,24,12,5,2,1,1,1,34,73,52,29,12,5,2,1,1,1,55,146,115,62,29,
%U A096669 12,5,2,1,1,1,89,293,251,140,70,29,12,5,2,1,1,1,144,585,542,321,156,70
%N A096669 Rectangular array T(n,k) read by antidiagonals; generating function of row n is 1/F(n,x), where F(n,x) is the polynomial 1 - x - x^2 - 2*x^3 -...- F(n+1)*x^n, and F(n+1) is the (n+1)st Fibonacci number, for n=0,1,2,...
%e A096669 Rows begin:
%e A096669 1 1 1 1 1 ... = A000012, with g.f. 1/(1-x)
%e A096669 1 1 2 3 5 ... = A000045, with g.f. 1/(1-x-x^2)
%e A096669 1 1 2 5 9 ... = A077947, with g.f. 1/(1-x-x^2-2*x^3)
%Y A096669 Cf. A000045, A096670. Rows converge to A000129.
%Y A096669 Adjacent sequences: A096666 A096667 A096668 this_sequence A096670 A096671 A096672
%Y A096669 Sequence in context: A054123 A119269 A129713 this_sequence A096591 A115568 A072909
%K A096669 nonn,tabl
%O A096669 1,8
%A A096669 Clark Kimberling (ck6(AT)evansville.edu), Jul 03 2004

%I A096591
%S A096591 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,5,5,2,1,1,1,8,10,5,2,1,1,1,15,22,14,5,
%T A096591 2,1,1,1,28,47,34,14,5,2,1,1,1,61,113,88,42,14,5,2,1,1,1,133,269,223,
%U A096591 116,42,14,5,2,1,1,1,328,705,609,333,132,42,14,5,2,1,1,1,807,1843,1660
%N A096591 Triangle T, read by rows, such that the matrix square shifts T one place diagonally left and upward, with T(n,0)=T(n,n)=1 for n>=0.
%C A096591 Column with index 1 forms the row sums shift right. The convergent of the rows in reverse order is the Catalan sequence (A000108).
%F A096591 T(n, k) = Sum_{i=0..n-1} T(n-2, i)*T(i, k-1) for n>1 and k>0; T(n, 0)=T(n, n)=1.
%e A096591 Rows begin:
%e A096591 [1],
%e A096591 [1,1],
%e A096591 [1,1,1],
%e A096591 [1,2,1,1],
%e A096591 [1,3,2,1,1],
%e A096591 [1,5,5,2,1,1],
%e A096591 [1,8,10,5,2,1,1],
%e A096591 [1,15,22,14,5,2,1,1],
%e A096591 [1,28,47,34,14,5,2,1,1],
%e A096591 [1,61,113,88,42,14,5,2,1,1],
%e A096591 [1,133,269,223,116,42,14,5,2,1,1],...
%e A096591 The matrix square of T is given by:
%e A096591 [1],
%e A096591 [2,1],
%e A096591 [3,2,1],
%e A096591 [5,5,2,1],
%e A096591 [8,10,5,2,1],
%e A096591 [15,22,14,5,2,1],
%e A096591 [28,47,34,14,5,2,1],
%e A096591 [61,113,88,42,14,5,2,1],
%e A096591 [133,269,223,116,42,14,5,2,1],...
%e A096591 which equals T shift one place diagonally left and upward.
%o A096591 (PARI) T(n,k)=if(n<k|k<0,0,if(n==k|k==0,1,sum(i=0,n-1,T(n-2,i)*T(i,k-1));))
%Y A096591 Cf. A096592, A000108.
%Y A096591 Adjacent sequences: A096588 A096589 A096590 this_sequence A096592 A096593 A096594
%Y A096591 Sequence in context: A119269 A129713 A096669 this_sequence A115568 A072909 A095691
%K A096591 nonn,tabl
%O A096591 0,8
%A A096591 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 28 2004

%I A115568
%S A115568 0,1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,1,
%T A115568 1,1,1,3,1,1,1,2,2,1,1,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,1,1,1,2,1,1,1,3,1,
%U A115568 1,2,2,1,1,1,1,1,2,1,1,1,3,1,2,1,2,1,1,1,5,1,2,2,2
%N A115568 Maximal Fibonacci exponent in prime factorization of n.
%Y A115568 Adjacent sequences: A115565 A115566 A115567 this_sequence A115569 A115570 A115571
%Y A115568 Sequence in context: A129713 A096669 A096591 this_sequence A072909 A095691 A067029
%K A115568 easy,nonn
%O A115568 0,5
%A A115568 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Mar 11 2006; revised Mar 13 2006

%I A072909
%S A072909 1,1,2,1,1,1,3,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,2,1,1,1,2,
%T A072909 1,1,1,2,1,1,1,3,2,1,1,4,3,2,1,2,1,2,1,2,1,1,1,2,1,1,3,2,1,1,1,2,1,1,1,
%U A072909 2,1,1,3,2,1,1,1,3,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,4,3,2,1,1,1,2,1,1
%N A072909 Least k>0 such that n+k is squarefree.
%F A072909 It seems that sum(k=1, n, a(k) ) is asymptotic to C*n with C=1.5...
%o A072909 (PARI) a(n)=if(n<0,0,s=1; while(issquarefree(n+s)==0,s++); s)
%Y A072909 Adjacent sequences: A072906 A072907 A072908 this_sequence A072910 A072911 A072912
%Y A072909 Sequence in context: A096669 A096591 A115568 this_sequence A095691 A067029 A087179
%K A072909 easy,nonn
%O A072909 1,3
%A A072909 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 10 2002

%I A095691
%S A095691 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,4,1,1,1,
%T A095691 4,1,1,1,3,1,1,1,2,2,1,1,3,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,4,1,1,1,2,1,1,
%U A095691 1,6,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1,3,1,2,1,2,1,1,1,4,1,2,2,4,1,1,1,3,1
%N A095691 Multiplicative with a(p^e) = A000720(e)+1.
%Y A095691 Cf. A095683.
%Y A095691 Adjacent sequences: A095688 A095689 A095690 this_sequence A095692 A095693 A095694
%Y A095691 Sequence in context: A096591 A115568 A072909 this_sequence A067029 A087179 A088388
%K A095691 mult,nonn
%O A095691 1,4
%A A095691 Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 06 2004

%I A067029
%S A067029 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,1,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,1,
%T A067029 2,1,1,1,3,1,1,1,2,2,1,1,4,2,1,1,2,1,1,1,3,1,1,1,2,1,1,2,6,1,1,1,2,1,1,
%U A067029 1,3,1,1,1,2,1,1,1,4,4,1,1,2,1,1,1,3,1,1
%N A067029 Exponent of least prime factor in prime factorization of n, a(1)=0.
%C A067029 Even bisection is A001511: a(2n) = A007814(n) + 1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jan 31 2004
%H A067029 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b067029.txt">Table of n, a(n) for n=1..10000</a>
%F A067029 A028233(n) = A020639(n)^a(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 13 2006
%e A067029 a(18) = a(2^1 * 3^2) = 1.
%Y A067029 Cf. A051903, A020639, A028233, A034684.
%Y A067029 Adjacent sequences: A067026 A067027 A067028 this_sequence A067030 A067031 A067032
%Y A067029 Sequence in context: A115568 A072909 A095691 this_sequence A087179 A088388 A070013
%K A067029 nonn,nice
%O A067029 1,4
%A A067029 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 17 2002

%I A087179
%S A087179 1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,1,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,1,4,
%T A087179 1,1,1,3,1,1,1,2,2,1,1,4,2,1,1,2,1,1,1,3,1,1,1,2,1,1,2,6,1,1,1,2,1,1,1,
%U A087179 9,1,1,1,2,1,1,1,4,4,1,1,2,1,1,1,3,1,1,1,2,1,1,1,5,1,1,2,4,1,1,1,3,1,1
%N A087179 a(n) = ((...(x1^x2)^x3)^x4)^...) where x1,x2,... are the exponents in the prime factorization of n.
%H A087179 MathMedics, <a href="http://www.sosmath.com/tables/factor/factor.html">The Prime Factorization of the First 1000 Integers</a>
%e A087179 a(108)=a((2^2)*(3^3))=2^3=8
%e A087179 a(75)=a((3^1)*(5^2))=1^2=1
%Y A087179 Adjacent sequences: A087176 A087177 A087178 this_sequence A087180 A087181 A087182
%Y A087179 Sequence in context: A072909 A095691 A067029 this_sequence A088388 A070013 A070014
%K A087179 nonn
%O A087179 2,3
%A A087179 Sam Alexander (amnalexander(AT)yahoo.com), Oct 19 2003

%I A088388
%S A088388 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,1,1,1,1,3,2,1,3,1,1,1,1,5,1,1,1,
%T A088388 2,1,1,1,3,1,1,1,1,2,1,1,4,2,2,1,1,1,3,1,3,1,1,1,1,1,1,2,6,1,1,1,1,1,1,
%U A088388 1,2,1,1,2,1,1,1,1,4,4,1,1,1,1,1,1,1,1,2,1,1,1,1,1,5,1,2,1,2,1,1
%N A088388 Exponent of largest prime power factor of n, a(1)=0.
%C A088388 A034699(n) = A088387(n)^a(n).
%Y A088388 Adjacent sequences: A088385 A088386 A088387 this_sequence A088389 A088390 A088391
%Y A088388 Sequence in context: A095691 A067029 A087179 this_sequence A070013 A070014 A051903
%K A088388 nonn
%O A088388 1,4
%A A088388 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 28 2003

%I A070013
%S A070013 1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,2,1,3,2,1,1,1,5,1,1,1,2,
%T A070013 1,1,1,2,1,1,1,2,2,1,1,3,2,2,1,2,1,2,1,2,1,1,1,1,1,1,2,6,1,1,1,2,1,1,1,
%U A070013 3,1,1,2,2,1,1,1,3,4,1,1,1,1,1,1,2,1,1,1,2,1,1,1,3,1,2,2,2,1,1,1,2,1,1
%N A070013 Number of prime factors of n divided by the number of n's distinct prime factors (rounded).
%C A070013 a(n) is the rounded average of the exponents in the prime factorization of n.
%F A070013 a(n) = round(bigomega(n)/omega(n)) for n>=2.
%e A070013 a(12)=2 because 12=2^2 * 3^1 and round(bigomega(12)/omega(12))=round((2+1)/2)=2. a(36)=2 because 36=2^2 * 3^2 and round(bigomega(36)/omega(36))=round((2+2)/2)=2. a(60)=1 because 60=2^2 * 3^1 * 5^1 and round(bigomega(60)/omega(60))= round((2+1+1)/3)=1. 36 is in A067340. 12 and 60 are in A070011.
%o A070013 (PARI) v=[]; for(n=2,150,v=concat(v,round(bigomega(n)/omega(n)))); v
%Y A070013 Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is integer before rounding), A070011 (ratio is not an integer), A070012 (floor of ratio), A070014 (ceiling of ratio), A046660 (bigomega(n)-omega(n)).
%Y A070013 Adjacent sequences: A070010 A070011 A070012 this_sequence A070014 A070015 A070016
%Y A070013 Sequence in context: A067029 A087179 A088388 this_sequence A070014 A051903 A072411
%K A070013 nonn
%O A070013 2,3
%A A070013 Rick L. Shepherd (rshepherd2(AT)hotmail.com), Apr 11 2002

%I A070014
%S A070014 1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,2,1,3,2,1,1,1,5,1,1,1,2,
%T A070014 1,1,1,2,1,1,1,2,2,1,1,3,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,6,1,1,1,2,1,1,1,
%U A070014 3,1,1,2,2,1,1,1,3,4,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,1,2,2,2,1,1,1,2,1,1
%N A070014 Ceiling of number of prime factors of n divided by the number of n's distinct prime factors.
%C A070014 a(n) is the ceiling of the average of the exponents in the prime factorization of n.
%F A070014 a(n) = ceiling(bigomega(n)/omega(n)) for n>=2.
%e A070014 a(12)=2 because 12=2^2 * 3^1 and ceil(bigomega(12)/omega(12))=ceil((2+1)/2)=2. a(36)=2 because 36=2^2 * 3^2 and ceil(bigomega(36)/omega(36))=ceil((2+2)/2)=2. a(60)=1 because 60=2^2 * 3^1 * 5^1 and ceil(bigomega(60)/omega(60))= ceil((2+1+1)/3)=2. 36 is in A067340. 12 and 60 are in A070011.
%o A070014 (PARI) v=[]; for(n=2,150,v=concat(v,ceil(bigomega(n)/omega(n)))); v
%Y A070014 Cf. A001221 (omega(n)), A001222 (bigomega(n)), A067340 (ratio is integer before ceil is applied), A070011 (ratio is not an integer), A070012 (floor of ratio), A070013 (ratio rounded), A046660 (bigomega(n)-omega(n)).
%Y A070014 Adjacent sequences: A070011 A070012 A070013 this_sequence A070015 A070016 A070017
%Y A070014 Sequence in context: A087179 A088388 A070013 this_sequence A051903 A072411 A091050
%K A070014 nonn
%O A070014 2,3
%A A070014 Rick L. Shepherd (rshepherd2(AT)hotmail.com), Apr 11 2002

%I A051903
%S A051903 0,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,1,
%T A051903 2,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,1,1,
%U A051903 1,3,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1,3,1,2,1,2,1,1,1,5,1,2,2,2,1,1,1,3,1
%N A051903 Maximal exponent in prime factorization of n.
%C A051903 Smallest number of factors of all factorizations of n into square-free numbers, see also A128651, A001055. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 30 2007
%H A051903 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b051903.txt">Table of n, a(n) for n=1..10000</a>
%H A051903 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NivensConstant.html">Niven's Constant</a>
%e A051903 For n = 72 = 2^3*3^2, a(72) = Max(exponents) = Max(3,2) = 3.
%t A051903 Table[If[n == 1, 0, Max @@ Last /@ FactorInteger[n]], {n, 100}] (*Chandler*)
%Y A051903 Cf. A005361, A008479, A051904, A052409, A091050.
%Y A051903 Cf. A129132.
%Y A051903 Adjacent sequences: A051900 A051901 A051902 this_sequence A051904 A051905 A051906
%Y A051903 Sequence in context: A088388 A070013 A070014 this_sequence A072411 A091050 A005361
%K A051903 nonn,easy
%O A051903 1,4
%A A051903 Labos E. (labos(AT)ana.sote.hu), Dec 16 1999

%I A072411
%S A072411 1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,1,2,
%T A072411 1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,1,1,1,
%U A072411 6,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1,3,1,2,1,2,1,1,1,5,1,2,2,2,1,1,1,3,1,1
%N A072411 LCM of exponents in prime factorization of n.
%e A072411 n=288=2.2.2.2.2.3.3; LCM[5,2]=10; Product[5,2]=10, Max[5,2]=5; n=180=2.2.3.3.5; LCM[2,2,1]=2; Product[2,2,1]= 4; Max[2,2,1]=2; It deviates both from Max of exponents(A051903) and product of exponents (A005361).
%t A072411 Table[LCM @@ Last /@ FactorInteger[n], {n, 2, 100}] (*Chandler*)
%Y A072411 Cf. A005361, A051903, A051904, A052409, A072412-A072414.
%Y A072411 Adjacent sequences: A072408 A072409 A072410 this_sequence A072412 A072413 A072414
%Y A072411 Sequence in context: A070013 A070014 A051903 this_sequence A091050 A005361 A008479
%K A072411 nonn
%O A072411 2,3
%A A072411 Labos E. (labos(AT)ana.sote.hu), Jun 17 2002

%I A091050
%S A091050 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,1,
%T A091050 4,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,1,1,
%U A091050 1,5,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1,3,1,2,1,2,1,1,1,5,1,2,2,4,1,1
%N A091050 Number of divisors of n that are perfect powers.
%C A091050 a(n)=1 iff n is square-free: a(A005117(n))=1, a(A013929(n))>1;
%C A091050 a(p^k)=k for p prime, k>0: a(A000961(n))=A025474(n);
%C A091050 not the same as A005361: a(72)=5 <> A005361(72)=6.
%H A091050 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>
%H A091050 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>
%F A091050 a(n) = A073093(n)-A001221(n) = A001222(n)-A001221(n)+1. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 28 2007
%e A091050 Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108}, a(108) =
%e A091050 #{1^2, 2^2, 3^2, 3^3, 6^2} = 5.
%Y A091050 Cf. A091051, A001597, A000005.
%Y A091050 Adjacent sequences: A091047 A091048 A091049 this_sequence A091051 A091052 A091053
%Y A091050 Sequence in context: A070014 A051903 A072411 this_sequence A005361 A008479 A107345
%K A091050 nonn
%O A091050 1,4
%A A091050 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 15 2003

%I A005361 M0063
%S A005361 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,
%T A005361 3,2,1,1,1,5,1,1,1,4,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,
%U A005361 1,3,1,3,1,1,1,2,1,1,2,6,1,1,1,2,1,1,1,6,1,1,2,2,1,1
%N A005361 Product of exponents of prime factorization of n.
%C A005361 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375 =3*5^3 both have prime signature (3,1).
%C A005361 a(n) is the number of nilpotents elements in the ring Z/nZ. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 02 2004
%D A005361 Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681.
%H A005361 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b005361.txt">Table of n, a(n) for n=1..10000</a>
%F A005361 n = Product (p_j^k_j) -> a(n) = Product (k_j). Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/zeta(6s).
%F A005361 Multiplicative with a(p^e) = e. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%t A005361 Prepend[ Array[ Times @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
%o A005361 (PARI) for(n=1,100,print1(prod(i=1,omega(n), component(component(factor(n),2),i)),","))
%Y A005361 Cf. A000005, A052306. a(p^k)=A000027=n. a(A002110)=A000012=1.
%Y A005361 Adjacent sequences: A005358 A005359 A005360 this_sequence A005362 A005363 A005364
%Y A005361 Sequence in context: A051903 A072411 A091050 this_sequence A008479 A107345 A000688
%K A005361 nonn,easy,nice,mult
%O A005361 1,4
%A A005361 Jeffrey Shallit, Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A008479
%S A008479 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,3,1,2,1,1,1,4,2,1,3,2,1,1,1,5,
%T A008479 1,1,1,5,1,1,1,3,1,1,1,2,2,1,1,6,2,4,1,2,1,7,1,3,1,1,1,2,1,1,2,6,
%U A008479 1,1,1,2,1,1,1,8,1,1,3,2,1,1,1,5,4,1,1,2,1,1,1,3,1,3,1,2,1,1,1,9
%N A008479 Number of numbers <= n with same prime factors as n.
%D A008479 Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681. (Incorrect!)
%D A008479 Problem 5735, Amer. Math. Monthly, 97 (1990), 937.
%H A008479 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b008479.txt">Table of n, a(n) for n=1..10000</a>
%t A008479 PkTbl=Prepend[ Array[ Times @@ First[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ];1+Array[ Count[ Take[ PkTbl, #-1 ], PkTbl[ [ # ] ] ]&, Length[ PkTbl ] ]
%Y A008479 Cf. A005361, A007947.
%Y A008479 Adjacent sequences: A008476 A008477 A008478 this_sequence A008480 A008481 A008482
%Y A008479 Sequence in context: A072411 A091050 A005361 this_sequence A107345 A000688 A038538
%K A008479 nonn,easy
%O A008479 1,4
%A A008479 Jeffrey Shallit, Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A107345
%S A107345 1,1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,3,3,2,3,1,1,1,3,2,1,1,2,1,1,1,5,4,6,
%T A107345 3,6,3,3,2,6,3,1,1,3,1,1,1,4,3,3,2,3,1,1,1,3,2,1,1,2,1,1,1,6,5,10,4,10,
%U A107345 6,6,3,10,6,4,3,6,3,3,2,10,6,4,3,4,1,1,1,6,3,1,1,3,1,1,1,5,4,6,3,6,3,3
%N A107345 From the binary representation of n: binomial(number of zeros, number of blocks of contiguous zeros).
%C A107345 a(n) = binomial(A023416(n), A087116(n)); a(A003754(n)) = 1.
%C A107345 First occurrence of k: 1, 4, 8, 16, 32, 34, 128, 256, 512, 66, 2048, 4096, 8192, 16384, 130, 65536, 131072, 262144, 524288, 266, 258, ..., . k must occur by 2^k. - Robert G. Wilson v.
%C A107345 Record values: 1, 4, 8, 16, 32, 34, 66, 130, 258, 514, 522, 1026, 1034, 2058, 4106, 4138, 8202, 8234, 16394, 16426, 32810, 65578, 65706, 131114, 131242, 262186, 262314, 524458, 1048746, 1049258, 2097322, 2097834, 4194474, 4194986, 8389290, 8391338, ..., . - Robert G. Wilson v.
%H A107345 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bi.html#binary">Index entries for sequences related to binary expansion of n</a>
%t A107345 f[n_] := Block[{id = IntegerDigits[n, 2]}, Binomial[ Count[id, 0], Floor[(Length@ Split@ id + 1)/2]]]; Table[f@n, {n, 0, 102}] (* Robert G. Wilson v, (rgwv@rgwv.com), Apr 01 2008 *)
%Y A107345 Cf. A007088, A107782, A136277.
%Y A107345 Adjacent sequences: A107342 A107343 A107344 this_sequence A107346 A107347 A107348
%Y A107345 Sequence in context: A091050 A005361 A008479 this_sequence A000688 A038538 A088529
%K A107345 nonn
%O A107345 0,5
%A A107345 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 23 2005

%I A000688 M0064 N0020
%S A000688 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,5,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,7,1,1,1,4,
%T A000688 1,1,1,3,1,1,1,2,2,1,1,5,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,11,1,1,1,2,1,1,1,
%U A000688 6,1,1,2,2,1,1,1,5,5,1,1,2,1,1,1,3,1,2,1,2,1,1,1,7,1,2,2,4,1,1,1,3,1,1,1
%N A000688 Number of factorizations of n into prime powers greater than 1; number of Abelian groups of order n.
%C A000688 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
%C A000688 Also number of rings with n elements that are the direct product of fields; these are the commutative rings with n elements having no nilpotents; likewise the commutative rings where for every element x there is a k > 0 such that x^{k+1} = x. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 20 2006
%C A000688 Range is A033637.
%D A000688 P. Erdos and G. Szekeres, Ueber die Anzahl der Abelschen Gruppen gegebener Ordnung und ueber ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
%D A000688 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
%D A000688 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.
%D A000688 H.-E. Richert, Ueber die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr. 56 (1952) 21-32.
%D A000688 J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
%H A000688 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000688.txt">Table of n, a(n) for n = 1..10000</a>
%H A000688 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/abel/abel.html">Abelian Group Enumeration Constants</a>
%H A000688 B. Horvat, G. Jaklic and T. Pisanski, <a href="http://arXiv.org/abs/math.CO/0503183">On the number of Hamiltonian groups</a>
%H A000688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelianGroup.html">Link to a section of The World of Mathematics.</a>
%H A000688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Link to a section of The World of Mathematics.</a>
%H A000688 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KroneckerDecompositionTheorem.html">Link to a section of The World of Mathematics.</a>
%H A000688 <a href="http://www.research.att.com/~njas/sequences/Sindx_Gre.html#groups">Index entries for sequences related to groups</a>
%H A000688 <a href="http://www.research.att.com/~njas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000688 a(p^k) = number of partitions of k; a(mn)=a(m)a(n) if (m, n)=1.
%F A000688 Multiplicative with a(p^e) = A000041(e). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%p A000688 with(combinat): readlib(ifactors): for n from 1 to 120 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,ans): od: # from James A. Sellers Dec 07 2000
%t A000688 f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* Robert G. Wilson v Sep 22 2006 *)
%Y A000688 Cf. A000001, A000041, A000961, A001055, A034382, A046054, A046055, A046056, A050360.
%Y A000688 Cf. A055653.
%Y A000688 Adjacent sequences: A000685 A000686 A000687 this_sequence A000689 A000690 A000691
%Y A000688 Sequence in context: A005361 A008479 A107345 this_sequence A038538 A088529 A136565
%K A000688 nonn,core,easy,nice,mult
%O A000688 1,4
%A A000688 njas

%I A038538
%S A038538 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,6,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,8,1,
%T A038538 1,1,4,1,1,1,3,1,1,1,2,2,1,1,6,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,13,1,1,
%U A038538 1,2,1,1,1,6,1,1,2,2,1,1,1,6,6,1,1,2,1,1,1,3,1,2,1,2,1,1,1,8,1,2,2
%N A038538 Number of semisimple rings with n elements.
%C A038538 Enumeration uses Wedderburn-Artin theorem and fact that a finite division ring is a field.
%C A038538 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375 =3*5^3 both have prime signature (3,1).
%D A038538 T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag.
%F A038538 Dirichlet generating function: sum( a(n) / n^s, n=1..infinity) = product ( 1/(1-p^(-r*m^2*s)), r= 1..infinity, m=1..infinity, p prime ) = product ( zeta(k*s)^A046951(n), k=1..infinity).
%o A038538 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X-X^2))[n] (from R. Stephan)
%Y A038538 Cf. A027623, A046951, A052305. a(p^k)=A004101. a(A002110)=1.
%Y A038538 Adjacent sequences: A038535 A038536 A038537 this_sequence A038539 A038540 A038541
%Y A038538 Sequence in context: A008479 A107345 A000688 this_sequence A088529 A136565 A086291
%K A038538 nonn,nice,mult
%O A038538 1,4
%A A038538 Paolo Dominici (pl.dm(AT)libero.it)

%I A088529
%S A088529 1,1,2,1,1,1,3,2,1,1,3,1,1,1,4,1,3,1,3,1,1,1,2,2,1,3,3,1,1,1,5,1,1,1,2,
%T A088529 1,1,1,2,1,1,1,3,3,1,1,5,2,3,1,3,1,2,1,2,1,1,1,4,1,1,3,6,1,1,1,3,1,1,1,
%U A088529 5,1,1,3,3,1,1,1,5,4,1,1,4,1,1,1,2,1,4,1,3,1,1,1,3,1,3,3,2
%N A088529 Numerator of Bigomega(n)/Omega(n).
%F A088529 Let B = number of prime divisors of n with multiplicity, O = number of distinct prime divisors of n. Then a(n) = numerator of B/O.
%e A088529 S(4!) = bigomega(4!) / omega(4!) = 4/2 = 2
%o A088529 (PARI) for(x=2,100,y=bigomega(x)/omega(x);print1(numerator(y)","))
%Y A088529 Cf. A088530.
%Y A088529 Adjacent sequences: A088526 A088527 A088528 this_sequence A088530 A088531 A088532
%Y A088529 Sequence in context: A107345 A000688 A038538 this_sequence A136565 A086291 A016442
%K A088529 nonn,frac
%O A088529 2,3
%A A088529 Cino Hilliard (hillcino368(AT)gmail.com), Nov 16 2003

%I A136565
%S A136565 0,1,1,2,1,1,1,3,2,1,1,3,1,1,1,4,1,3,1,3,1,1,1,4,2,1,3,3,1,1,1,5,1,1,1,
%T A136565 2,1,1,1,4,1,1,1,3,3,1,1,5,2,3
%N A136565 a(n) = sum of the distinct values making up the exponents in the prime-factorization of n.
%e A136565 120 = 2^3 * 3^1 * 5^1. The exponents of the prime factorization are therefore 3,1,1. The distinct values which equal these exponents are 1 and 3. So a(120) = 1+3 = 4.
%Y A136565 Cf. A071625, A136566, A136568.
%Y A136565 Adjacent sequences: A136562 A136563 A136564 this_sequence A136566 A136567 A136568
%Y A136565 Sequence in context: A000688 A038538 A088529 this_sequence A086291 A016442 A076360
%K A136565 more,nonn
%O A136565 1,4
%A A136565 Leroy Quet (qq-quet(AT)mindspring.com), Jan 07 2008

%I A086291
%S A086291 0,1,1,2,1,1,1,3,2,1,2,1,1,4,2,2,1,3,2,3,2,1,5,1,2,3,1,2,4,2,2,3,3,2,2,
%T A086291 6,1,3,2,4,4,2,2,5,2,2,1,3,4,3,3,2,7,3,2,4,2,2,5,4,3,2,2,3,6,2,3,1,3,5,
%U A086291 2,4,5,2,3,2,8,3,3,5,2,2,2,6,4,4,3,2,3,3,3,7,3,4,4,2,4,2,6,2
%N A086291 Maximal exponent in prime factorization of 7-smooth numbers.
%C A086291 a(n) = A051903(A002473(n));
%C A086291 A086290(n) <= a(n) <= A086289(n).
%Y A086291 Adjacent sequences: A086288 A086289 A086290 this_sequence A086292 A086293 A086294
%Y A086291 Sequence in context: A038538 A088529 A136565 this_sequence A016442 A076360 A089398
%K A086291 nonn
%O A086291 1,4
%A A086291 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 15 2003

%I A016442
%S A016442 2,1,1,1,3,2,1,3,1,7,1,4,8,4,10,1,2,1,9,1,5,4,2,4,5,2,3,
%T A016442 123,10,12,4,1,1,1,1,1,2,2,9,4,1,4,11,2,1,8,2,2,3,3,11,
%U A016442 16,2,2,3,7,1,1,1,95,5,34,1,2,4,1,2,3,1,1,1,37,6,38,1,26
%N A016442 Continued fraction for ln(14).
%Y A016442 Adjacent sequences: A016439 A016440 A016441 this_sequence A016443 A016444 A016445
%Y A016442 Sequence in context: A088529 A136565 A086291 this_sequence A076360 A089398 A047040
%K A016442 nonn,cofr
%O A016442 1,1
%A A016442 njas

%I A076360
%S A076360 0,1,0,2,1,1,1,3,2,1,3,6,1,1,1,4,3,8,3,4,1,2,5,3,2,1,1,4,5,3,3,6,3,1,
%T A076360 3,9,1,5,1,3,5,3,3,0,4,5,7,12,0,8,5,6,5,1,5,1,3,5,9,12,1,5,4,6,5,0,3,0,5,0,
%U A076360 9,20,1,1,6,0,5,1,7,10,3,5,9,16,5,5,9,3,9,16,3,4,1,8,9,10,3,6,0,9,5,1,5,1,1
%V A076360 0,-1,0,-2,1,-1,1,-3,-2,-1,3,-6,1,1,1,-4,3,-8,3,-4,-1,2,5,-3,-2,1,1,-4,5,-3,3,-6,3,1,
%W A076360 3,-9,1,5,1,-3,5,-3,3,0,-4,5,7,-12,0,-8,5,-6,5,1,5,1,3,5,9,-12,1,5,-4,-6,5,0,3,0,5,0,
%X A076360 9,-20,1,1,-6,0,5,1,7,-10,-3,5,9,-16,5,5,9,3,9,-16,3,4,1,8,9,-10,3,-6,0,-9,5,1,5,1,-1
%N A076360 a(n)=Commutator[sigma,tau](n)=d0[d1[w]]-d1[d0[w]], where d0()=number of, d1()=sum of divisors of n.
%t A076360 d0[x_] := DivisorSigma[0, x] d1[x_] := DivisorSigma[1, x] Table[d0[d1[w]]-d1[d0[w]], {w, 1, 128}]
%Y A076360 Cf. A000005, A000203.
%Y A076360 Adjacent sequences: A076357 A076358 A076359 this_sequence A076361 A076362 A076363
%Y A076360 Sequence in context: A136565 A086291 A016442 this_sequence A089398 A047040 A047020
%K A076360 easy,sign
%O A076360 1,4
%A A076360 Labos E. (labos(AT)ana.sote.hu), Oct 08 2002

%I A089398
%S A089398 1,0,2,1,1,1,3,2,2,0,3,2,2,2,4,3,3,1,2,2,2,2,4,3,3,1,4,3,3,3,5,4,4,2,3,
%T A089398 1,2,2,4,3,3,1,4,3,3,3,5,4,4,2,3,3,3,3,5,4,4,2,5,4,4,4,6,5,5,3,4,2,1,2,
%U A089398 4,3,3,1,4,3,3,3,5,4,4,2,3,3,3,3,5,4,4,2,5,4,4,4,6,5,5,3,4,2,3,3,5,4,4
%N A089398 a(n) = n-th column sum of binary digits of k*2^(k-1), where summation is over all k>=1, without carrying from columns sums that may exceed 2.
%C A089398 sum(k=1,n, a(k)*2^(k-1)) = 2^A089399(n)+1 for n>2, with a(1)=a(2)=1.
%C A089398 Row sums of triangular arrays in A103588 and in A103589. - Philippe DELEHAM, Apr 04 2005
%C A089398 a(k) = 0 for k = 2, 10, 2058, 2058 + 2^2059, ..., that is, for k = A034797(n) - 1, n>=2. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2007
%F A089398 a(2^n)=n-1 (for n>0), a(2^n-1)=n (for n>0), a(2^n+1)=n-1 (for n>1), a(2^n-k)=n-A089400(k) (for n>k>0), a(2^n+k)=n-A089401(k) (for n>k>0), where sequences have limits: A089400={0, 2, 2, 2, 1, 4, 2, 2, 1, 3, 3, ...}, and A089401={1, 1, 3, 2, 4, 5, 6, 5, 7, 8, 11, 9, ...},
%e A089398 Binary expansions of k*2^(k-1), with bits in ascending order by powers of 2, are:
%e A089398 1
%e A089398 001
%e A089398 0011
%e A089398 000001
%e A089398 0000101
%e A089398 00000011
%e A089398 000000111
%e A089398 00000000001
%e A089398 000000001001
%e A089398 0000000000101
%e A089398 00000000001101
%e A089398 000000000000011
%e A089398 0000000000001011
%e A089398 .................
%e A089398 Giving column sums:
%e A089398 10211132203222433...
%t A089398 f[n_] := Block[{lg = Floor[Log[2, n]] + 1}, Sum[ Join[ Reverse[ IntegerDigits[n - i + 1, 2]], {0}][[i]], {i, lg}]]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Mar 26 2005)
%Y A089398 Cf. A089399, A089400, A089401.
%Y A089398 Adjacent sequences: A089395 A089396 A089397 this_sequence A089399 A089400 A089401
%Y A089398 Sequence in context: A086291 A016442 A076360 this_sequence A047040 A047020 A127832
%K A089398 base,nonn
%O A089398 1,3
%A A089398 Paul D. Hanna (pauldhanna(AT)juno.com), Oct 30 2003

%I A047040
%S A047040 1,1,1,1,1,1,1,2,1,1,1,3,2,2,1,1,4,5,2,3,1,1,5,9,5,5,4,1,1,6,14,14,10,
%T A047040 9,5,1,1,7,20,28,14,9,14,6,1,1,8,27,48,42,14,23,20,7,1,1,9,35,75,90,56,
%U A047040 23,43,27,8,1,1,10,44,110,165,146,56,66,70,35
%N A047040 Array T read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=3x/4 unless x=0 or x=h.
%e A047040 Diagonal (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,1,1}; ...
%Y A047040 Mirror image of A047050.
%Y A047040 Adjacent sequences: A047037 A047038 A047039 this_sequence A047041 A047042 A047043
%Y A047040 Sequence in context: A016442 A076360 A089398 this_sequence A047020 A127832 A107249
%K A047040 nonn,tabl
%O A047040 0,8
%A A047040 Clark Kimberling (ck6(AT)evansville.edu)

%I A047020
%S A047020 1,1,1,1,1,1,1,2,1,1,1,3,2,2,1,1,4,5,4,3,1,1,5,9,5,3,4,1,1,6,14,14,5,7,
%T A047020 5,1,1,7,20,28,19,7,12,6,1,1,8,27,48,47,19,19,18,7,1,1,9,35,75,95,66,
%U A047020 38,37,25,8,1,1,10,44,110,170,161,66,37,62,33,9
%N A047020 Array T read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=2x/3 unless x=0 or x=h.
%e A047020 Diagonals (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,1,1}; ...
%Y A047020 Adjacent sequences: A047017 A047018 A047019 this_sequence A047021 A047022 A047023
%Y A047020 Sequence in context: A076360 A089398 A047040 this_sequence A127832 A107249 A062842
%K A047020 nonn,tabl
%O A047020 0,8
%A A047020 Clark Kimberling (ck6(AT)evansville.edu)

%I A127832
%S A127832 0,1,1,1,1,1,1,2,1,1,1,3,2,2,1,1,4,5,4,3,1,1,5,9,9,7,4,1,1,6,14,18,9,11,
%T A127832 5,1,1,7,20,32,27,11,16,6,1,1,8,27,52,59,27,27,22,7,1,1,9,35,79,111,86,
%U A127832 54,49,29,8,1,1,10,44,114,190,197,140,103,78,37,9,1,1,11,54,158,304,387
%N A127832 Array T by antidiagonals: for h>=0 and k>=0, T(h,k)=number of UR paths from (0,0) to (h,k) that touch the line y=2x/3 only at lattice points. A UR path is a path of steps of length 1 each directed up or right.
%e A127832 Northwest corner:
%e A127832 0 1 1 1 1 1
%e A127832 1 1 2 3 4 5
%e A127832 1 1 2 5 9 14
%e A127832 1 2 4 9 18 32
%e A127832 1 3 7 9 27 59
%e A127832 The 4 UR paths for T(3,2) are RRRUU, RRURU, UURRR, URURR. These
%e A127832 paths touch the line y=2x/3 only at the lattice points (0,0) and (3,2).
%Y A127832 Cf. A047110, A127833.
%Y A127832 Adjacent sequences: A127829 A127830 A127831 this_sequence A127833 A127834 A127835
%Y A127832 Sequence in context: A089398 A047040 A047020 this_sequence A107249 A062842 A126805
%K A127832 nonn,tabl
%O A127832 1,8
%A A127832 Clark Kimberling (ck6(AT)evansville.edu), Feb 01 2007

%I A107249
%S A107249 1,1,1,1,1,1,2,1,1,1,3,2,2,1,1,6,3,3,2,1,1,10,6,6,3,3,1,1,20,10,10,6,4,
%T A107249 3,1,1,35,20,20,10,10,4,4,1,1,70,35,35,20,15,10,5,4,1,1,126,70,70,35,35,
%U A107249 15,15,5,5,1,1,252,126,126,70,56,35,21,15,6,5,1,1,462,252,252,126,126
%N A107249 A number triangle with repeated columns of binomial coefficients.
%C A107249 Reversal of A096921. Row sums are A026010. Diagonal sums are 107250.
%F A107249 Number triangle T(n, k)=C(floor((2n-k-2)/2), floor((n-k)/2))
%e A107249 Triangle begins
%e A107249 1;
%e A107249 1,1;
%e A107249 1,1,1;
%e A107249 2,1,1,1;
%e A107249 3,2,2,1,1;
%e A107249 6,3,3,2,1,1;
%e A107249 10,6,6,3,3,1,1;
%Y A107249 Adjacent sequences: A107246 A107247 A107248 this_sequence A107250 A107251 A107252
%Y A107249 Sequence in context: A047040 A047020 A127832 this_sequence A062842 A126805 A110963
%K A107249 easy,nonn,tabl
%O A107249 0,7
%A A107249 Paul Barry (pbarry(AT)wit.ie), May 14 2005

%I A062842
%S A062842 0,1,1,2,1,1,1,3,2,2,1,2,1,1,1,4,3,3,2,3,2,2,1,3,2,2,3,2,2,2,1,5,4,4,3,
%T A062842 4,3,3,2,4,3,3,2,3,2,2,1,4,3,3,2,3,2,3,2,3,2,2,1,2,1,1,2,6,5,5,4,5,4,4,
%U A062842 3,5,4,4,3,4,3,3,2,5,4,4,3,4,3,3,3,4,3,3,2,3,2,2,1,5,4,4,3,4
%N A062842 Maximum number of zeros in the representation of n in any base.
%e A062842 a[10]=2 since 10 in base 2 is 1010, containing 2 zeros
%Y A062842 Adjacent sequences: A062839 A062840 A062841 this_sequence A062843 A062844 A062845
%Y A062842 Sequence in context: A047020 A127832 A107249 this_sequence A126805 A110963 A106348
%K A062842 base,easy,nonn
%O A062842 1,4
%A A062842 Erich Friedman (efriedma(AT)stetson.edu), Jul 21 2001

%I A126805
%S A126805 1,1,1,1,2,1,1,1,3,2,2,1,2,2,4,2,3,2,3,2,1,2,3,3,1,2,2,3,1,2,2,2,2,4,2,
%T A126805 2,2,1,4,3,4,2,2,1,2,3,2,2,3,2,3,2,2,2,1,3,4,2,4,2,5,2,2,3,2,3,3,2,4,3,
%U A126805 3,5,3,3,2,3,2,3,2,2,3,2,3,1,2,2,2,2,4,3,4,3,1,2,4,3,3,2,3,2,2,5,3,3,2
%N A126805 "Class-" (or "class-minus") number of prime(n) according to the Erdos-Selfridge classification of primes.
%C A126805 This gives the "class-" number as opposed to the "class+" number. Not to be confused with the "class-number" of quadratic form theory.
%C A126805 a(n)=1 if A000040(n) is in A005109, a(n)=2 if A000040(n) is in A005110, a(n)=3 if A000040(n) is in A005111 etc.
%F A126805 a(n) = max { a(p)+1 ; prime(p) is > 3 and divides prime(n)-1 } union { 1 } - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 16 2007
%p A126805 a := proc(n) option remember; local p,pf,e,res; if isprime(n) then pf := ifactors(n-1)[2]; res := 1; for e from 1 to nops(pf) do p := op(1,op(e,pf)); if p > 3 then res := max(res,a(p)+1); fi; od; RETURN(res); else -1; fi; end: for n from 1 to 180 do printf("%d, ",a(ithprime(n))); end:
%o A126805 (PARI) A126805(n) = { if( n>0, n=-prime(n)); if(( n=factor(-1-n)[,1] ) & n[ #n]>3, vecsort( vector( #n, i, A126805(-n[i]) ))[ #n]+1, 1) } - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 16 2007
%Y A126805 Cf. A005109, A005110, A005111, A005112, A081424, A081425.
%Y A126805 Cf. A081640, A129247, A129248, A056637.
%Y A126805 Adjacent sequences: A126802 A126803 A126804 this_sequence A126806 A126807 A126808
%Y A126805 Sequence in context: A127832 A107249 A062842 this_sequence A110963 A106348 A029332
%K A126805 easy,nonn
%O A126805 1,5
%A A126805 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 23 2007

%I A110963
%S A110963 1,1,1,1,2,1,1,1,3,2,2,1,4,1,1,1,5,3,3,2,6,2,2,1,7,4,4,1,8,1,1,1,9,5,5,
%T A110963 3,10,3,3,2,11,6,6,2,12,2,2,1,13,7,7,4,14,4,4
%N A110963 Fractalisation of a fractal: of the Kimberling's sequence beginning with 1.
%C A110963 Self-descriptive sequence: even terms are the sequence itself, odd terms (the skeleton of this sequence) are the terms of the Kimberling's sequence beginning with 1. Also: -a(4n) = the natural numbers -a(4n+1)= the Kimberling's sequence (beginning with 1) -a(4n+2)= the Kimberling's sequence (beginning with 1) -a(4n+3)= the sequence itself -a(8n+1)=a(8n+2)= the natural numbers.
%H A110963 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">Fractal sequences</a>.
%F A110963 a(2n+1)=a(n)=a(4n+3) = terms of the sequence itself. a(2n)=a(4n+1)=a(4n+2) = terms of Kimberling's sequence (beginning with 1). a(4n)=a(8n+1)=a(8n+2)= n.
%Y A110963 Cf. A110812, A110779, A110766. Equals A110962 + 1.
%Y A110963 Adjacent sequences: A110960 A110961 A110962 this_sequence A110964 A110965 A110966
%Y A110963 Sequence in context: A107249 A062842 A126805 this_sequence A106348 A029332 A134431
%K A110963 base,easy,nonn,uned
%O A110963 0,5
%A A110963 Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Sep 26 2005

%I A106348
%S A106348 1,0,2,1,1,1,3,2,2,2,2,2,2,2,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,4,4,4,4,
%T A106348 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,5,5,5,5,5,5,5,
%U A106348 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A106348 Partial sums of a generalized Fredholm-Rueppel sequence.
%C A106348 Partial sums of A106347.
%F A106348 a(2^(n+1)-2)=n+1; a(k)=n-1 for 2^n-2<k<2^(n+1)-2
%Y A106348 Adjacent sequences: A106345 A106346 A106347 this_sequence A106349 A106350 A106351
%Y A106348 Sequence in context: A062842 A126805 A110963 this_sequence A029332 A134431 A070879
%K A106348 easy,nonn
%O A106348 0,3
%A A106348 Paul Barry (pbarry(AT)wit.ie), Apr 29 2005

%I A029332
%S A029332 1,0,0,0,1,1,0,1,2,1,1,1,3,2,2,3,4,3,3,4,6,5,5,6,8,7,7,
%T A029332 8,11,10,10,11,14,13,13,15,18,17,17,19,23,21,22,24,28,27,
%U A029332 27,30,34,33,34,36,41,40,41,44,49,48,49,52,58,57,58,62
%N A029332 Expansion of 1/((1-x^4)(1-x^5)(1-x^7)(1-x^8)).
%Y A029332 Adjacent sequences: A029329 A029330 A029331 this_sequence A029333 A029334 A029335
%Y A029332 Sequence in context: A126805 A110963 A106348 this_sequence A134431 A070879 A125644
%K A029332 nonn,easy
%O A029332 0,9
%A A029332 njas

%I A134431
%S A134431 1,1,1,1,1,1,2,1,1,1,3,2,2,6,1,1,1,3,3,4,8,8,6,6,24,1,1,1,3,3,5,10,10,
%T A134431 14,14,36,30,30,24,24,120,1,1,1,3,3,5,11,12,16,22,44,44,66,60,78,174,
%U A134431 168,144,144,120,120,720,1,1,1,3,3,5,11,13,18,24,52,52,80,98,120,234
%N A134431 Triangle read by rows: T(n,k) is the number of arrangements of the set {1,2,...,n} in which the sum of the entries is equal to k (n>=0, k>=0; to n=0 there corresponds the empty set).
%C A134431 Row n has 1 + n(n+1)/2 terms (n>=0). Row sums yield the arrangement numbers (A000522). T(n,n(n+1)/2)=n!. Sum(k*T(n,k),k=0..n(n+1)/2)=A134432(n)
%F A134431 The row generating polynomials P[n](t) are equal to Q[n](t,1), where the polynomials Q[n](t,x) are defined by Q[0]=1 and Q[n]=Q[n-1] + xt^n diff(xQ[n-1], x). [Q[n](t,x) is the bivariate generating polynomial of the arrangements of {1,2,...,n}, where t (x) marks the sum (number) of the entries; for example, Q[2](t,x)=1+tx + t^2*x + 2t^3*x^2, corresponding to: empty, 1, 2, 12, and 21, respectively.]
%e A134431 T(4,7)=8 because we have 34,43, and the six permutations of {1,2,4}.
%e A134431 Triangle starts:
%e A134431 1;
%e A134431 1,1;
%e A134431 1,1,1,2;
%e A134431 1,1,1,3,2,2,6;
%e A134431 1,1,1,3,3,4,8,8,6,6,24;
%p A134431 Q[0]:=1: for n to 7 do Q[n]:=sort(simplify(Q[n-1]+t^n*x*(diff(x*Q[n-1],x))), t) end do: for n from 0 to 7 do P[n]:=sort(subs(x=1,Q[n])) end do: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..(1/2)*n*(n+1)) end do; # yields sequence in triangular form
%Y A134431 Cf. A000522, A134432.
%Y A134431 Adjacent sequences: A134428 A134429 A134430 this_sequence A134432 A134433 A134434
%Y A134431 Sequence in context: A110963 A106348 A029332 this_sequence A070879 A125644 A048821
%K A134431 nonn,tabf
%O A134431 0,7
%A A134431 Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 16 2007

%I A070879
%S A070879 1,1,1,1,2,1,1,1,3,2,3,1,2,1,1,1,4,3,5,2,5,3,4,1,3,2,3,1,2,1,1,1,5,4,7,
%T A070879 3,8,5,7,2,7,5,8,3,7,4,5,1,4,3,5,2,5,3,4,1,3,2,3,1,2,1,1,1,6,5,9,4,11,7,
%U A070879 10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4,9,5,6,1,5,4,7,3,8
%N A070879 Stern's diatomic array read by r