The Database of Integer Sequences, Part 2 

     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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(start)



%I A089798
%S A089798 1,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,
%T A089798 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A089798 0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0
%V A089798 1,0,-2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,
%W A089798 0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,
%X A089798 0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0
%N A089798 Expansion of Jacobi theta function theta_4(q^2).
%D A089798 I. J. Zucker, "Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums." J. Phys. A: Math. Gen. 23, 117-132, 1990.
%H A089798 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%Y A089798 Cf. A002448.
%Y A089798 Adjacent sequences: A089795 A089796 A089797 this_sequence A089799 A089800 A089801
%Y A089798 Sequence in context: A117371 A117370 A112053 this_sequence A070536 A030201 A055668
%K A089798 sign
%O A089798 0,3
%A A089798 Eric Weisstein (eric(AT)weisstein.com), Nov 12, 2003

%I A070536
%S A070536 1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,4,0,
%T A070536 10,0,0,0,4,0,0,2,0,0,2,0,0,0,0,0,6,0,0,0,6,0,6,0,0,2,0,0,2,0,18,4,0,0,
%U A070536 8,10,0,0,0,0,2,0,20,4,0,0,0,0,0,2,24,0,10,0,0,2,10,0,10,0,12,0,0,0,4
%N A070536 Number of terms in n-th cyclotomic polynomial minus largest prime factor of n; a(1)=1 by convention.
%C A070536 When (as at n=105) coefficients are not equal 1 or -1 then terms in C[n,x] are counted with multiplicity.
%e A070536 n=21: Cyclotomic[21,x]=1-x+x^3-x^4+x^6-x^8+x^9-x^11+x^12 has 9 terms while largest prime factor of 21 is 7
%Y A070536 a(n)=A051664[n]-A06530[n].
%Y A070536 Cf. A006530, A051664, A070537, A070776.
%Y A070536 Adjacent sequences: A070533 A070534 A070535 this_sequence A070537 A070538 A070539
%Y A070536 Sequence in context: A117370 A112053 A089798 this_sequence A030201 A055668 A045839
%K A070536 nonn
%O A070536 1,15
%A A070536 Labos E. (labos(AT)ana.sote.hu), May 03 2002

%I A030201
%S A030201 0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,
%T A030201 0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,
%U A030201 0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0
%V A030201 0,1,0,0,-1,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,
%W A030201 0,1,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,-2,0,0,0,0,0,1,0,0,0,0,
%X A030201 0,0,0,0,0,0,0,0,0,0,-1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0
%N A030201 Expansion of eta(q^3)*eta(q^21).
%D A030201 M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
%Y A030201 Adjacent sequences: A030198 A030199 A030200 this_sequence A030202 A030203 A030204
%Y A030201 Sequence in context: A112053 A089798 A070536 this_sequence A055668 A045839 A000086
%K A030201 sign
%O A030201 0,38
%A A030201 njas

%I A055668
%S A055668 0,0,0,1,1,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,
%T A055668 0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,
%U A055668 0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0
%N A055668 Number of inequivalent Eisenstein-Jacobi primes of norm n.
%C A055668 These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.
%C A055668 Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-omega, +-omega^2).
%D A055668 R. K. Guy, Unsolved Problems in Number Theory, A16.
%D A055668 L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.
%F A055668 a(n) = 2 if n is a prime = 1 (mod 6); a(n) = 1 if n = 3 or n = p^2 where p is a prime = 2 (mod 3); a(n) = 0 otherwise. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 05 2006
%e A055668 There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.
%Y A055668 Cf. A055664-A055667, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.
%Y A055668 Adjacent sequences: A055665 A055666 A055667 this_sequence A055669 A055670 A055671
%Y A055668 Sequence in context: A089798 A070536 A030201 this_sequence A045839 A000086 A045838
%K A055668 nonn,easy,nice
%O A055668 0,8
%A A055668 njas, Jun 09 2000
%E A055668 More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 05 2006

%I A045839
%S A045839 0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,1,0,
%T A045839 0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,
%U A045839 0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,0
%N A045839 A005929/2.
%Y A045839 Adjacent sequences: A045836 A045837 A045838 this_sequence A045840 A045841 A045842
%Y A045839 Sequence in context: A070536 A030201 A055668 this_sequence A000086 A045838 A045837
%K A045839 nonn
%O A045839 0,8
%A A045839 njas

%I A000086
%S A000086 1,0,1,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,
%T A000086 0,2,0,2,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,2,0,0,0,
%U A000086 0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,2,0,0,0,2,0,0,0,0,0,2,0,0
%N A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).
%C A000086 Number of elliptic points of order 3 for GAMMA_0 (n).
%C A000086 Equivalently, number of fixed points of GAMMA_0 (n) of type rho.
%C A000086 Values are 0 or a power of 2.
%D A000086 Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations. J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
%D A000086 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
%D A000086 G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).
%H A000086 Christian G. Bower, <a href="http://www.research.att.com/~njas/sequences/b000086.txt">Table of n, a(n) for n=1..2000</a>
%F A000086 Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%p A000086 with(numtheory); A000086 := proc (n) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end: (Gene Smith, May 22 2006)
%t A000086 Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
%o A000086 (PARI) a(n)=if(n<1,0,sum(x=0,n-1,(x^2-x+1)%n==0))
%o A000086 (PARI) a(n)=if(n<1,0,direuler(p=2,n,if(p==3,1+X,if(p%3==2,1,(1+X)/(1-X))))[n])
%Y A000086 Cf. A000089, A000091, A001616, A014683.
%Y A000086 Adjacent sequences: A000083 A000084 A000085 this_sequence A000087 A000088 A000089
%Y A000086 Sequence in context: A030201 A055668 A045839 this_sequence A045838 A045837 A126825
%K A000086 nonn,easy,nice,mult
%O A000086 1,7
%A A000086 njas

%I A045838
%S A045838 0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,
%T A045838 3,0,0,0,0,0,1,0,2,0,0,0,4,0,2,0,0,0,2,0,0,0,0,0,2,0,4,
%U A045838 0,0,0,5,0,2,0,0,0,2,0,0,0,0,0,2,0,4,0,1,0,3
%N A045838 A005871/2.
%Y A045838 Adjacent sequences: A045835 A045836 A045837 this_sequence A045839 A045840 A045841
%Y A045838 Sequence in context: A055668 A045839 A000086 this_sequence A045837 A126825 A045833
%K A045838 nonn
%O A045838 0,10
%A A045838 njas

%I A045837
%S A045837 0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,1,0,2,0,0,0,2,0,
%T A045837 1,0,0,0,2,0,2,0,0,0,2,0,2,0,0,0,1,0,2,0,0,0,2,0,4,0,0,
%U A045837 0,2,0,2,0,0,0,2,0,4,0,0,0,3,0,4,0,0,0,0,0,3,0,0,0,2
%N A045837 A005888/2.
%Y A045837 Adjacent sequences: A045834 A045835 A045836 this_sequence A045838 A045839 A045840
%Y A045837 Sequence in context: A045839 A000086 A045838 this_sequence A126825 A045833 A117896
%K A045837 nonn
%O A045837 0,10
%A A045837 njas

%I A126825
%S A126825 1,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,
%T A126825 0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,
%U A126825 0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,0,0,2,0,0
%N A126825 Ramanujan numbers (A000594) read mod 3.
%D A126825 H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients, of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
%Y A126825 Cf. A000594.
%Y A126825 Adjacent sequences: A126822 A126823 A126824 this_sequence A126826 A126827 A126828
%Y A126825 Sequence in context: A000086 A045838 A045837 this_sequence A045833 A117896 A132976
%K A126825 nonn
%O A126825 1,7
%A A126825 njas, Feb 25 2007

%I A045833
%S A045833 0,1,0,0,1,0,0,2,0,0,0,0,0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,
%T A045833 0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,
%U A045833 0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,2,0,0,1,0,0,2,0
%N A045833 Expansion of eta(q^9)^3/eta(q^3) in powers of q.
%F A045833 Euler transform of period 9 sequence [0, 0, 1, 0, 0, 1, 0, 0, -2, ...]. - Michael Somos, May 25 2005
%F A045833 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2*w-2u*w^2-v^3. - Michael Somos May 25 2005
%F A045833 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1*u3^2+u1*u6^2-u1*u3*u6-u2^2*u3. - Michael Somos May 25 2005
%F A045833 a(3n)=a(3n+2)=0. a(3n+1)=A033687(n). a(6n+1)=A097195(n). 3*a(n)=A033685(n).
%F A045833 Multiplicative with a(3^e) = 0 if e>0, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3). - Michael Somos May 25 2005
%F A045833 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u2*u3^2+2*u2*u3*u6+4*u2*u6^2-u1^2*u6. - Michael Somos May 25 2005
%e A045833 q +q^4 +2*q^7 +2*q^13 +q^16 +2*q^19 +q^25 +2*q^28 +2*q^31 +...
%o A045833 (PARI) {a(n)=local(A,p,e); if(n<0, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p!=3, if(p%3==1, e+1, !(e%2))))))} /* Michael Somos May 25 2005 */
%o A045833 (PARI) {a(n)=local(A); if((n<1)|(n%3!=1), 0, n=(n-1)/3; A=x*O(x^n); polcoeff( eta(x^3+A)^3/eta(x+A), n))} /* Michael Somos May 25 2005 */
%Y A045833 Adjacent sequences: A045830 A045831 A045832 this_sequence A045834 A045835 A045836
%Y A045833 Sequence in context: A045838 A045837 A126825 this_sequence A117896 A132976 A028649
%K A045833 nonn,mult
%O A045833 0,8
%A A045833 njas

%I A117896
%S A117896 0,1,0,0,2,0,0,0,0,0,2,0,0,1,1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,
%T A117896 1,0,0,0,0,1,0,0,0,1,2,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,
%U A117896 0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0
%N A117896 Number of perfect powers between consecutive squares n^2 and (n+1)^2.
%C A117896 a(n)=2 only 14 times for n<2^63. What is the least n such that a(n)=3? Is a(n) bounded?
%e A117896 a(5)=2 because powers 27 and 32 are between 25 and 36.
%t A117896 nMax=150^2; lst={}; log2Max=Ceiling[Log[2,nMax]]; bases=Table[2,{log2Max}]; powers=bases^Range[log2Max]; powers[[1]]=Infinity; cnt=0; While[nextPP=Min[powers]; nextPP <= nMax, pos=Flatten[Position[powers,nextPP]]; If[MemberQ[pos,2], AppendTo[lst,cnt]; cnt=0, cnt++ ]; Do[k=pos[[i]]; bases[[k]]++; powers[[k]]=bases[[k]]^k, {i,Length[pos]}]]; lst
%Y A117896 Cf. A001597 (perfect powers), A014085 (primes between squares).
%Y A117896 Adjacent sequences: A117893 A117894 A117895 this_sequence A117897 A117898 A117899
%Y A117896 Sequence in context: A045837 A126825 A045833 this_sequence A132976 A028649 A097798
%K A117896 nonn
%O A117896 1,5
%A A117896 T. D. Noe (noe(AT)sspectra.com), Mar 31 2006

%I A132976
%S A132976 1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2,0,0,3,0,0,1,0,
%T A132976 0,4,0,0,4,0,0,1,0,0,4,0,0,6,0,0,1,0,0,5,0,0,8,0,0,1,0,0,8,0,0,10,0,0,2,
%U A132976 0,0,11,0,0,14,0,0,4,0,0,14,0,0,19,0,0,4,0,0,17,0,0,24,0,0,4,0,0,23
%V A132976 1,-1,0,-1,0,0,1,0,0,1,0,0,-1,0,0,0,0,0,1,0,0,-2,0,0,0,0,0,2,0,0,-3,0,0,1,0,0,4,0,0,-4,
%W A132976 0,0,1,0,0,4,0,0,-6,0,0,1,0,0,5,0,0,-8,0,0,1,0,0,8,0,0,-10,0,0,2,0,0,11,0,0,-14,0,0,4,
%X A132976 0,0,14,0,0,-19,0,0,4,0,0,17,0,0,-24,0,0,4,0,0,23
%N A132976 Expansion of (1/q) * psi(-q) / psi(-q^9) in powers of q where psi() is a Ramanujan theta function.
%F A132976 Expansion of eta(q) * eta(q^4) * eta(q^18) / (eta(q^2) * eta(q^9) * eta(q^36)) in powers of q.
%F A132976 Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, ...].
%F A132976 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u3 * u6 - (u1 + u2 + u1*u2) * (u3 + u6 + 3).
%F A132976 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 / f(t) where q = exp(2 pi i t).
%F A132976 a(3*n+1) = 0. a(3*n) = 0 unless n=0.
%e A132976 1/q - 1 - q^2 + q^5 + q^8 - q^11 + q^17 - 2*q^20 + 2*q^26 - 3*q^29 + ...
%o A132976 (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x*O(x^n) ; polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A) / ( eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A) ), n))}
%Y A132976 A062244(n) = a(3*n-1). Convolution inverse of A132975.
%Y A132976 Adjacent sequences: A132973 A132974 A132975 this_sequence A132977 A132978 A132979
%Y A132976 Sequence in context: A126825 A045833 A117896 this_sequence A028649 A097798 A065205
%K A132976 sign
%O A132976 -1,22
%A A132976 Michael Somos, Sep 07 2007

%I A028649
%S A028649 1,0,2,0,0,0,0,0,2,0,0,4,0,0,0,4,0,0,2,0,0,0,0,4,0,0,0,
%T A028649 0,0,0,0,0,2,0,0,4,0,0,0,0,0,0,2,0,4,0,0,0,0,0,6,4,0,0,
%U A028649 0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,2,0,4,0,0,0,0,0,0
%N A028649 Expansion of (theta_3(z)*theta_3(21z)+theta_2(z)*theta_2(21z)).
%Y A028649 Adjacent sequences: A028646 A028647 A028648 this_sequence A028650 A028651 A028652
%Y A028649 Sequence in context: A045833 A117896 A132976 this_sequence A097798 A065205 A036272
%K A028649 nonn
%O A028649 0,3
%A A028649 njas

%I A097798
%S A097798 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,0,0,2,0,1,0,0,0,
%T A097798 4,0,1,0,2,0,4,0,2,0,0,0,7,0,2,0,2,0,8,0,5,0,2,0,14,0,4,0,4,0,14,0,8,0,
%U A097798 5,0,23,0,9,0,9,0,26,0,18,0,9,0,38,0,16,0,17,0,46,0,29,0,19,0,65,0,32,0
%N A097798 Number of partitions of n into abundant numbers.
%H A097798 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>
%H A097798 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Partition.html">Partition</a>
%Y A097798 Cf. A005101, A000041, A097800, A097797, A097796, A097795.
%Y A097798 Adjacent sequences: A097795 A097796 A097797 this_sequence A097799 A097800 A097801
%Y A097798 Sequence in context: A117896 A132976 A028649 this_sequence A065205 A036272 A083339
%K A097798 nonn
%O A097798 1,24
%A A097798 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 25 2004

%I A065205
%S A065205 0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,1,0,0,0,5,0,0,0,1,0,3,0,0,0,0,0,
%T A065205 7,0,0,0,3,0,2,0,0,0,0,0,10,0,0,0,0,0,3,0,2,0,0,0,34,0,0,0,0,0,2,0,0,0,
%U A065205 0,0,31,0,0,0,0,0,1,0,6,0,0,0,25,0,0,0,1,0,23,0,0,0,0,0,21,0,0,0,2
%N A065205 Number of subsets of proper divisors of n that sum to n.
%C A065205 Deficient and weird numbers have a(n)=0, perfect numbers and others (see A064771) have a(n)=1.
%H A065205 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b065205.txt">Table of n, a(n) for n=1..1000</a>
%e A065205 {1,4,5,10} is the only subset of proper divisors of 20 that sum to 20, so a(20)=1.
%Y A065205 Cf. A064771.
%Y A065205 a(n) = A033630(n) - 1.
%Y A065205 Adjacent sequences: A065202 A065203 A065204 this_sequence A065206 A065207 A065208
%Y A065205 Sequence in context: A132976 A028649 A097798 this_sequence A036272 A083339 A133827
%K A065205 nonn
%O A065205 1,12
%A A065205 Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 19 2001
%E A065205 More terms and additional comments from Jud McCranie (j.mccranie(AT)comcast.net), Oct 21 2001

%I A036272
%S A036272 1,2,0,0,0,0,0,2,0,2,0,0,0,2,4,0,2,0,2,2,0,0,2,2,0,0,0,8,0,8,2,
%T A036272 4,0,4,4,2,0,2,4,4,0,6,0,8,10,8,6,0,0,2,4,4,4,0,4,0,2,0,6,4,6,8,
%U A036272 0,8,2,4,4,6,0,0,0,2,2,0,0,2,0,2,6,0,0,4,2,0,0,2,2,0,6,4,0,0,0
%N A036272 Absolute values of differences of absolute values of second differences of primes.
%F A036272 |DIFF(|DIFF(DIFF(primes))|)|.
%Y A036272 Cf. A036262, A036263.
%Y A036272 Adjacent sequences: A036269 A036270 A036271 this_sequence A036273 A036274 A036275
%Y A036272 Sequence in context: A028649 A097798 A065205 this_sequence A083339 A133827 A028633
%K A036272 nonn
%O A036272 0,2
%A A036272 njas

%I A083339
%S A083339 0,0,0,1,0,1,0,0,1,1,0,0,0,1,2,0,0,0,0,0,2,1,0,0,1,1,1,0,0,0,0,0,2,1,2,
%T A083339 0,0,1,2,0,0,0,0,0,2,1,0,0,1,0,2,0,0,0,2,0,2,1,0,0,0,1,2,0,2,0,0,0,2,0,
%U A083339 0,0,0,1,2,0,2,0,0,0,1,1,0,0,2,1,2,0,0,0,2,0,2,1,2,0,0,0,2,0,0,0,0,0,3
%N A083339 Number of distinct prime factors of n that occur in prime-partitions confirming Goldbach's conjectures.
%H A083339 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture.</a>
%H A083339 <a href="http://www.research.att.com/~njas/sequences/Sindx_Go.html#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e A083339 n = 14 = 2*7 = 3+11=7+7, only one factor of 14 occurs: a(14)=1;
%e A083339 n = 15 = 3*5 = 2+2+11=3+5+7=5+5+5, both factors of 15 occur:
%e A083339 a(15)=2.
%Y A083339 Cf. A001221, A083338, A045917, A054860.
%Y A083339 Adjacent sequences: A083336 A083337 A083338 this_sequence A083340 A083341 A083342
%Y A083339 Sequence in context: A097798 A065205 A036272 this_sequence A133827 A028633 A074039
%K A083339 nonn
%O A083339 1,15
%A A083339 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 24 2003

%I A133827
%S A133827 1,0,0,1,1,2,0,0,0,0,0,2,1,0,2,0,0,0,2,0,0,2,0,0,1,0,2,0,0,0,0,1,0,2,0,
%T A133827 2,0,0,2,2,1,0,0,0,0,0,0,0,0,2,0,0,0,2,2,0,2,0,0,0,3,0,0,2,0,0,0,0,2,0,
%U A133827 0,0,0,0,2,2,0,0,0,0,2,2,0,0,1,0,0,1,0,2,0,0,0,0,0,2,2,0,2,0,0,2,0,2,0
%N A133827 Number of solutions to x + 7 * y = 2 * n in triangular numbers.
%C A133827 G.f. called omega(q) by Berkovich and Yesilyurt.
%H A133827 A. Berkovich and H. Yesilyurt, <a href="http://arxiv.org/abs/math.NT/0603150">New Identities For 7-cores with Prescribed BG-Rank</a>, page 3 equation (1.19)
%F A133827 Expansion of phi(q) * psi(q^7) - q * psi(q^2) * psi(q^14) in powers of q^2 where psi() is a Ramanujan theta function.
%F A133827 Expansion of psi(q^4) * phi(q^14) + q^3 * psi(q^28) * phi(q^2) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A133827 a(n) = b(2*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(7^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), b(p^e) = e + 1 if p == 1, 2, 4 (mod 7).
%F A133827 a(7*n+1) = a(7*n+2) = a(7*n+6) = 0. a(7*n+3) = a(n).
%e A133827 q + q^7 + q^9 + 2*q^11 + 2*q^23 + q^25 + 2*q^29 + 2*q^37 + 2*q^43 + q^49 + ...
%o A133827 (PARI) {a(n) = if( n<0, 0, n = 2*n+1; sumdiv(n, d, (d%2) * kronecker( -28, d)))}
%o A133827 (PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*n+1; A = factor(n); prod(k = 1, matsize(A) [1], if(p = A[k,1], e = A[k,2]; if(p == 2, 0, if( p == 7, 1, if( 1 == kronecker( -7, p), e + 1, !(e%2)) )))))}
%Y A133827 A035162(2*n+1) = a(n).
%Y A133827 Adjacent sequences: A133824 A133825 A133826 this_sequence A133828 A133829 A133830
%Y A133827 Sequence in context: A065205 A036272 A083339 this_sequence A028633 A074039 A047764
%K A133827 nonn
%O A133827 0,6
%A A133827 Michael Somos, Sep 25 2007

%I A028633
%S A028633 1,0,2,0,0,0,0,0,2,4,0,0,0,4,0,0,0,0,2,0,0,4,0,0,0,0,0,
%T A028633 0,0,0,0,0,2,4,2,0,4,0,0,0,0,0,4,0,0,0,0,0,0,4,2,0,4,0,
%U A028633 0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,0,0,4,0,0,0
%N A028633 Expansion of (theta_3(z)*theta_3(17z)+theta_2(z)*theta_2(17z)).
%Y A028633 Adjacent sequences: A028630 A028631 A028632 this_sequence A028634 A028635 A028636
%Y A028633 Sequence in context: A036272 A083339 A133827 this_sequence A074039 A047764 A022883
%K A028633 nonn
%O A028633 0,3
%A A028633 njas

%I A074039
%S A074039 0,0,1,0,2,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,
%T A074039 0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,
%U A074039 8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A074039 If (n, n+2) is the k-th twin prime pair then k else 0.
%C A074039 A001359(a(n)) = n, A006512(a(n)) = n+2.
%Y A074039 Cf. A074038, A049084.
%Y A074039 Adjacent sequences: A074036 A074037 A074038 this_sequence A074040 A074041 A074042
%Y A074039 Sequence in context: A083339 A133827 A028633 this_sequence A047764 A022883 A028833
%K A074039 nonn
%O A074039 1,5
%A A074039 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 13 2002

%I A047764
%S A047764 0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,
%T A047764 7,0,0,0,0,0,12,0,0,0,0,0,30,0,0,0,0,0,55,0,0,0,0,0,143,0,0,0,0,
%U A047764 0,273,0,0,0,0,0,728,0,0,0,0,0,1428,0,0,0,0,0,3876,0,0,0,0,0,7752
%N A047764 Number of dissectable polyhedra with symmetry of type Q.
%D A047764 L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
%F A047764 If n=6m+2 then A047749((n-2)/6), else 0.
%Y A047764 Cf. A027610.
%Y A047764 Adjacent sequences: A047761 A047762 A047763 this_sequence A047765 A047766 A047767
%Y A047764 Sequence in context: A133827 A028633 A074039 this_sequence A022883 A028833 A024943
%K A047764 nonn
%O A047764 1,20
%A A047764 njas

%I A022883
%S A022883 0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,1,0,
%T A022883 0,1,0,0,1,0,1,0,0,3,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,1,0,0,0,0,0,3,0,
%U A022883 1,0,0,2,0,0,1,0,0,1,0,1,0,0,3,0,0,0,0,4,0,0,0,0
%N A022883 The number of numbers [ [ ix ]jx ] that equal n, where i >= 1, j >= 1 and x=e.
%Y A022883 Adjacent sequences: A022880 A022881 A022882 this_sequence A022884 A022885 A022886
%Y A022883 Sequence in context: A028633 A074039 A047764 this_sequence A028833 A024943 A005929
%K A022883 nonn
%O A022883 1,21
%A A022883 Clark Kimberling (ck6(AT)evansville.edu)

%I A028833
%S A028833 0,0,0,0,0,0,2,0,0,0,0,0,3,1,0,0,0,0,2,0,3,2,1,0,0,0,0,1,2,0,4,2,1,1,
%T A028833 0,0,0,0,0,0,1,0,6,5,3,7,1,0,0,0,0,2,2,2,2,0,3,5,2,1,6,1,0,0,0,0,5,0,
%U A028833 4,3,4,0,4,3,2,7,2,1,1,0,0,0,0,0,2,6,0,3,2,0,5,4,6,10,1,1,8,1,0,0,0,0
%N A028833 Index of redundancy in period of continued fraction for sqrt(n).
%F A028833 Difference between A003285 and A028832.
%Y A028833 Adjacent sequences: A028830 A028831 A028832 this_sequence A028834 A028835 A028836
%Y A028833 Sequence in context: A074039 A047764 A022883 this_sequence A024943 A005929 A005871
%K A028833 nonn,nice
%O A028833 1,7
%A A028833 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A024943
%S A024943 0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1,
%T A024943 1,1,0,0,0,1,2,0,0,0,0,0,3,1,0,0,0,0,2,2,0,0,0,1,1,3,0,0,0,1,2,2,1,0,0,1,3,
%U A024943 2,1,0,0,1,4,3,1,0,0,0,3,5,1,0,0,0,3,5,2,0,0,1,3,5,4,0,0,1,4,5,3,1,0,1,5,8
%N A024943 Number of partitions of n into distinct 6k+1 primes.
%Y A024943 Adjacent sequences: A024940 A024941 A024942 this_sequence A024944 A024945 A024946
%Y A024943 Sequence in context: A047764 A022883 A028833 this_sequence A005929 A005871 A005888
%K A024943 nonn
%O A024943 1,44
%A A024943 Clark Kimberling (ck6(AT)evansville.edu)

%I A005929 M0006
%S A005929 0,2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,
%T A005929 0,0,4,0,0,0,0,0,4,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,
%U A005929 0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,4,0,0,0,0,0,4,0
%N A005929 Theta series of hexagonal net with respect to mid-point of edge.
%D A005929 N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
%p A005929 theta2:=x->add(2*x^((2*n-1)^2/4),n=1..200): phi1:=x->theta2(x^(1/4))*theta2(x^(3/4))/2: seq(coeff(convert(series(phi1(x^4)-phi1(x^12),x,200),polynom),x,n),n=0..200); - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 28 2008
%Y A005929 Adjacent sequences: A005926 A005927 A005928 this_sequence A005930 A005931 A005932
%Y A005929 Sequence in context: A022883 A028833 A024943 this_sequence A005871 A005888 A107499
%K A005929 easy,nonn
%O A005929 0,2
%A A005929 njas
%E A005929 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 28 2008

%I A005871 M0007
%S A005871 0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,8,0,0,0,0,0,6,0,0,0,0,0,2,0,4,
%T A005871 0,0,0,8,0,4,0,0,0,4,0,0,0,0,0,4,0,8,0,0,0,10,0,4,0,0,0,4,0,0,0,0,0,4,0,8,0,2,0,6
%N A005871 Theta series of hexagonal close-packing with respect to edge within layer.
%D A005871 N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
%Y A005871 Adjacent sequences: A005868 A005869 A005870 this_sequence A005872 A005873 A005874
%Y A005871 Sequence in context: A028833 A024943 A005929 this_sequence A005888 A107499 A123298
%K A005871 easy,nonn
%O A005871 0,4
%A A005871 njas

%I A005888 M0008
%S A005888 0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,2,0,4,0,0,0,4,0,2,0,0,0,4,0,4,0,
%T A005888 0,0,4,0,4,0,0,0,2,0,4,0,0,0,4,0,8,0,0,0,4,0,4,0,0,0,4,0,8,0,0,0,6,0,8,0,0,0,0,0,6,0,0,0,4
%N A005888 Theta series of hexagonal close-packing with respect to edge between layers.
%D A005888 N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
%Y A005888 Adjacent sequences: A005885 A005886 A005887 this_sequence A005889 A005890 A005891
%Y A005888 Sequence in context: A024943 A005929 A005871 this_sequence A107499 A123298 A110981
%K A005888 easy,nonn
%O A005888 0,4
%A A005888 njas

%I A107499
%S A107499 1,0,0,2,0,0,0,0,0,4,4,0,8,2,10,0,10,8
%N A107499 Coefficients of a certain theta series.
%C A107499 See reference for details.
%D A107499 W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.
%Y A107499 Adjacent sequences: A107496 A107497 A107498 this_sequence A107500 A107501 A107502
%Y A107499 Sequence in context: A005929 A005871 A005888 this_sequence A123298 A110981 A019261
%K A107499 nonn
%O A107499 0,4
%A A107499 njas, May 28 2005

%I A123298
%S A123298 0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,4,6,18,14,26,84,88,46,176,380,812,844,
%T A123298 1770,2164,5554,9244,25384,62092,68176,85762,304892,855072,1229050,
%U A123298 1805096
%N A123298 a(n) = number of incongruent restricted disjoint covering systems (IRDCS) of length n. An IRDCS of length n is a sequence of n integers s(1),...,s(n) with the property that if s(i)=m for some m then the other members of the sequence with value m are precisely s(i+km) for all k such that i+km is in [1,n], and further such that any integer appearing in the sequence appears at least twice. For example the two IRDCS of length n are 6,9,3,4,5,3,6,4,3,5,9 and its reverse.
%D A123298 Gerry Myerson, Jacky Poon and Jamie Simpson, Incongruent restricted disjoint covering systems, preprint.
%Y A123298 Adjacent sequences: A123295 A123296 A123297 this_sequence A123299 A123300 A123301
%Y A123298 Sequence in context: A005871 A005888 A107499 this_sequence A110981 A019261 A019222
%K A123298 nice,nonn
%O A123298 1,11
%A A123298 Jamie Simpson (simpson(AT)maths.curtin.edu.au), Jun 25 2007, corrected Sep 11 2007

%I A110981
%S A110981 1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,24,0,6,0,0,0,236,0,0,0,18,0,3768,
%T A110981 0,0,0,0,0,20384,0,0,0,7188,0,227784,0,186,480,0,0,1732448,0,237600,
%U A110981 0,630,0,16028160,0,306684,0,0,0,341521732,0,0,4896,0,0,1417919208
%N A110981 a(n) = the number of aperiodic subsets S of the n-th roots of 1 with zero sum (i.e. there is no r different from 1 such that r*S=S).
%C A110981 We count these subsets only modulo rotations (multiplication by a nontrivial root of unity).
%C A110981 A103314(n) = a(n)*n + 2^n - A001037(n)*n. Note that as soon as a(n)=0, we have simply A103314(n) = 2^n - A001037(n)*n. This makes it especially interesting to study those n for which a(n)=0. It is a surprising fact that the sequence of such n coincides with A102466.
%C A110981 Comment from Max Alekseyev, Jan 31 2008 (Start): Every subset of the set U(n) = { 1=r^0, r^1, ..., r^(n-1) } of the n-th power roots of 1 (where r is a fixed primitive root) defines a binary word w of the length n where the j-th bit is 1 iff the root r^j is included in the subset.
%C A110981 If d is the period of w with respect to cyclic rotations (thus d|n) then the periodic part of w uniquely defines some binary Lyndon word of the length d (see A001037). In turn, each binary Lyndon word of the length d, where d<n and d|n, corresponds to d distinct zero-sum subsets of U(n).
%C A110981 The binary Lyndon words of the length n are different in this respect: only some of them correspond to n distinct zero-sum subsets of U(n) while the others do not correspond to such subsets at all. A110981(n) gives the number of binary Lyndon words of the length n that correspond to zero-sum subsets of U(n). (End)
%H A110981 Max Alekseyev and M. F. Hasler, <a href="http://www.research.att.com/~njas/sequences/b110981.txt">Table of n, a(n) for n = 1..164</a>
%F A110981 a(n) = A001037(n) - A107847(n) ( = A001037(n) - (2^n - A103314(n))/n ). - M. F. Hasler, Jan 31 2008
%Y A110981 Cf. A103314, A001037, A107847.
%Y A110981 Adjacent sequences: A110978 A110979 A110980 this_sequence A110982 A110983 A110984
%Y A110981 Sequence in context: A005888 A107499 A123298 this_sequence A019261 A019222 A019141
%K A110981 nonn
%O A110981 1,12
%A A110981 Max Alekseyev (maxale(AT)gmail.com), Jan 20 2008
%E A110981 Additional comments from M. F. Hasler, Jan 31 2008

%I A019261
%S A019261 2,0,0,0,0,0,24,92,208,456,1052,2304,5424,14064,36760,94848,243504,
%T A019261 621372
%N A019261 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite VSV = VPI-7 Na26H6[ Zn16Si56O144 ]. 44 H2O.
%D A019261 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019261 G. Thimm, <a href="http://www.research.att.com/~njas/sequences/thimm1.html">Cycle sequences of crystal structures</a>
%Y A019261 Adjacent sequences: A019258 A019259 A019260 this_sequence A019262 A019263 A019264
%Y A019261 Sequence in context: A107499 A123298 A110981 this_sequence A019222 A019141 A086077
%K A019261 nonn
%O A019261 3,1
%A A019261 Georg Thimm (mgeorg(AT)ntu.edu.sg)

%I A019222
%S A019222 2,0,0,0,0,0,26,86,228,424,1040,2192,5632,13820,37462,94070,244632,
%T A019222 612400
%N A019222 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite RSN = RUB-17 K4Na12 [ Zn8Si28O72 ]. 18 H2O.
%D A019222 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019222 G. Thimm, <a href="http://www.research.att.com/~njas/sequences/thimm1.html">Cycle sequences of crystal structures</a>
%Y A019222 Adjacent sequences: A019219 A019220 A019221 this_sequence A019223 A019224 A019225
%Y A019222 Sequence in context: A123298 A110981 A019261 this_sequence A019141 A086077 A073345
%K A019222 nonn
%O A019222 3,1
%A A019222 Georg Thimm (mgeorg(AT)ntu.edu.sg)

%I A019141
%S A019141 2,0,0,0,0,0,28,80,248,392,1028,2068,5856,13680,38100,92984,245540,
%T A019141 603684
%N A019141 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite LOV = Lovdarite K4Na12 [ Be8Si28O72 ] . 18 H2O.
%D A019141 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019141 G. Thimm, <a href="http://www.research.att.com/~njas/sequences/thimm1.html">Cycle sequences of crystal structures</a>
%Y A019141 Adjacent sequences: A019138 A019139 A019140 this_sequence A019142 A019143 A019144
%Y A019141 Sequence in context: A110981 A019261 A019222 this_sequence A086077 A073345 A112765
%K A019141 nonn
%O A019141 3,1
%A A019141 Georg Thimm (mgeorg(AT)ntu.edu.sg)

%I A086077
%S A086077 0,0,0,1,0,0,0,0,1,0,0,2,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,1,0,1,0,
%T A086077 1,3,0,0,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,1,0,
%U A086077 0,1,1,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,1,1,2,0,0,0,0,0,0,1,1
%N A086077 Number of 6's in decimal expansion of triangular number n(n+1)/2.
%Y A086077 Cf. 0's A086071, 1's A086072, 2's A086073, 3's A086074, 4's A086075, 5's A086076, 7's A086078, 8's A086079, 9's A086080.
%Y A086077 Adjacent sequences: A086074 A086075 A086076 this_sequence A086078 A086079 A086080
%Y A086077 Sequence in context: A019261 A019222 A019141 this_sequence A073345 A112765 A105966
%K A086077 base,nonn
%O A086077 0,12
%A A086077 Jason Earls (jcearls(AT)cableone.net), Jul 08 2003

%I A073345
%S A073345 1,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,6,0,0,0,
%T A073345 0,0,0,0,6,8,0,0,0,0,0,0,0,4,20,0,0,0,0,0,0,0,0,1,40,16,0,0,0,0,0,0,0,
%U A073345 0,0,68,56,0,0,0,0,0,0,0,0,0,0,94,152,32,0,0,0,0,0,0,0,0,0,0,114,376
%N A073345 Table T(n,k) (listed antidiagonalwise in order T(0,0), T(1,0), T(0,1), T(2,0), T(1,1), ...) giving the number of rooted plane binary trees of size n and height k.
%D A073345 Luo Jian-Jin, Catalan numbers in the history of mathematics in China, in Combinatorics and Graph Theory, (Yap, Ku, Lloyd, Wang, Editors), World Scientific, River Edge, NJ, 1995.
%H A073345 H. Bottomley and A. Karttunen, <A HREF="http://www.research.att.com/~njas/sequences/A073345.txt">Notes concerning diagonals of the square arrays A073345 and A073346.</A>
%H A073345 Andrew Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/enumeration.html">Analytic methods in asymptotic enumeration</a>.
%F A073345 (See the Maple code below. Is there a nicer formula?)
%F A073345 This table was known to the Chinese mathematician Ming An-Tu, who gave the following recurrence in the 1730s. a(0, 0) = 1, a(n, k) = Sum[a(n-1, k-1-i)( 2*Sum[ a(j, i), {j, 0, n-2}]+a(n-1, i) ), {i, 0, k-1}]. - David Callan (callan(AT)stat.wisc.edu), Aug 17 2004
%F A073345 The generating function for row n, T_n(x):=Sum[T(n, k)x^k, k>=0], is given by T_n = a(n)-a(n-1) where a(n) is defined by the recurrence a(0)=0, a(1)=1, a(n) = 1 + x a(n-1)^2 for n>=2. - David Callan (callan(AT)stat.wisc.edu), Oct 08 2005
%e A073345 The top-left corner of this square array is
%e A073345 1 0 0 0 0 0 0 0 0 ...
%e A073345 0 1 0 0 0 0 0 0 0 ...
%e A073345 0 0 2 1 0 0 0 0 0 ...
%e A073345 0 0 0 4 6 6 4 1 0 ...
%e A073345 0 0 0 0 8 20 40 68 94 ...
%e A073345 E.g. we have A000108(3) = 5 binary trees built from 3 non-leaf (i.e. branching) nodes:
%e A073345 _______________________________3
%e A073345 ___\/__\/____\/__\/____________2
%e A073345 __\/____\/__\/____\/____\/_\/__1
%e A073345 _\/____\/____\/____\/____\./___0
%e A073345 The first four have height 3, and the last one has height 2, thus T(3,3) = 4, T(3,2) = 1, and T(3,any other value of k) = 0.
%p A073345 A073345 := n -> A073345bi(A025581(n), A002262(n));
%p A073345 A073345bi := proc(n,k) option remember; local i,j; if(0 = n) then if(0 = k) then RETURN(1); else RETURN(0); fi; fi; if(0 = k) then RETURN(0); fi; 2 * add(A073345bi(n-i-1,k-1) * add(A073345bi(i,j),j=0..(k-1)),i=0..floor((n-1)/2)) + 2 * add(A073345bi(n-i-1,k-1) * add(A073345bi(i,j),j=0..(k-2)),i=(floor((n-1)/2)+1)..(n-1)) - (`mod`(n,2))*(A073345bi(floor((n-1)/2),k-1)^2); end;
%p A073345 A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);
%p A073345 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
%t A073345 a[0, 0] = 1; a[n_, k_]/;k<n||k>2^n-1 := 0; a[n_, k_]/;1 <= n <= k <= 2^n-1 := a[n, k] = Sum[a[n-1, k-1-i](2Sum[ a[j, i], {j, 0, n-2}]+a[n-1, i]), {i, 0, k-1}]; Table[a[n, k], {n, 0, 9}, {k, 0, 9}]
%t A073345 (* or *) a[0] = 0; a[1] = 1; a[n_]/;n>=2 := a[n] = Expand[1 + x a[n-1]^2]; gfT[n_] := a[n]-a[n-1]; Map[CoefficientList[ #, x, 8]&, Table[gfT[n], {n, 9}]/.{x^i_/;i>=9 ->0}] (Callan)
%Y A073345 Variant: A073346. Column sums: A000108. Row sums: A001699.
%Y A073345 Diagonals: A073345(n, n) = A011782(n), A073345(n+3, n+2) = A014480(n), A073345(n+2, n) = A073773(n), A073345(n+3, n) = A073774(n) - Henry Bottomley (se16(AT)btinternet.com) and AK, see the attached notes.
%Y A073345 A073429 gives the upper triangular region of this array. Cf. also A065329, A001263.
%Y A073345 Adjacent sequences: A073342 A073343 A073344 this_sequence A073346 A073347 A073348
%Y A073345 Sequence in context: A019222 A019141 A086077 this_sequence A112765 A105966 A083915
%K A073345 nonn,tabl
%O A073345 0,13
%A A073345 Antti Karttunen Jul 31 2002

%I A112765
%S A112765 0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1,
%T A112765 0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,
%U A112765 0,0,0,0,2,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1
%N A112765 Exponent of highest power of 5 dividing n.
%C A112765 A027868 gives partial sums.
%F A112765 Totally additive with a(p) = 1 if p = 5, 0 otherwise.
%Y A112765 Cf. A007814, A007949, A112762, A022337.
%Y A112765 Cf. A122840.
%Y A112765 Adjacent sequences: A112762 A112763 A112764 this_sequence A112766 A112767 A112768
%Y A112765 Sequence in context: A019141 A086077 A073345 this_sequence A105966 A083915 A083892
%K A112765 nonn
%O A112765 1,25
%A A112765 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 18 2005

%I A105966
%S A105966 1,0,0,0,1,1,0,0,1,0,0,0,0,1,0,0,0,0,1,1,2,0,0,0,0,1,0,0,0,0,1,0,0,0,1,
%T A105966 1,0,0,1,0,0,0,0,1,0,0,0,0,1,1,2,0,0,0,0,1,0,0,0,0,1,0,0,0,1,1,0,0,1,0,
%U A105966 0,0,0,1,0,0,0,0,1,1,2,0,0,0,0,1,0,0,0,0,1,0,0,0,1,1,0,0,1,0,0
%V A105966 1,0,0,0,1,1,0,0,1,0,0,0,0,-1,0,0,0,0,1,1,2,0,0,0,0,-1,0,0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,
%W A105966 0,-1,0,0,0,0,1,1,2,0,0,0,0,-1,0,0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,0,-1,0,0,0,0,1,1,2,0,0,
%X A105966 0,0,-1,0,0,0,0,1,0,0,0,1,1,0,0,1,0,0
%N A105966 Expansion of A/B with A = (-1+x^15-x^10-x^9-x^8-2*x^5-x^4) and B = ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)*(x^4-x^3+x^2-x+1)*(x^8-x^7+x^5-x^4+x^3-x+1).
%C A105966 Sequence appears to be periodic with initial period (1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0). (Period 30).
%o A105966 Floretion Algebra Multiplication Program, FAMP Code: 2ibasefizrokseq[ + .5'i + .5'ii' - .5'ij' + .5'ik'], RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + 1 (internal program code). FizType: ChuRed.
%Y A105966 Adjacent sequences: A105963 A105964 A105965 this_sequence A105967 A105968 A105969
%Y A105966 Sequence in context: A086077 A073345 A112765 this_sequence A083915 A083892 A089814
%K A105966 sign
%O A105966 0,21
%A A105966 Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 28 2005

%I A083915
%S A083915 0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2,
%T A083915 0,0,0,0,1,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2,
%U A083915 0,0,0,0,4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2,0,0,0,0,4
%N A083915 Number of divisors of n that are congruent to 5 modulo 10.
%C A083915 a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
%Y A083915 Cf. A010879, A000005, A001227.
%Y A083915 Adjacent sequences: A083912 A083913 A083914 this_sequence A083916 A083917 A083918
%Y A083915 Sequence in context: A073345 A112765 A105966 this_sequence A083892 A089814 A063100
%K A083915 nonn
%O A083915 1,15
%A A083915 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 08 2003

%I A083892
%S A083892 0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2,
%T A083892 0,0,0,0,1,0,0,0,0,3,0,0,0,0,3,1,1,1,1,2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,2,
%U A083892 0,0,0,0,3,0,0,0,0,1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,1,0,0,0,0,3,0,1,0,1,4
%N A083892 Number of divisors of n with largest digit = 5 (base 10).
%C A083892 a(n) = A000005(n) - A083888(n) - A083889(n) - A083890(n) - A083891(n) - A083893(n) - A083894(n) - A083895(n) - A083896(n) = A083900(n) - A083899(n).
%e A083892 n=125, 3 of the 4 divisors of 125 have largest digit =5:
%e A083892 {5,25,125}, therefore a(125)=3.
%Y A083892 Cf. A054055, A000005, A083900.
%Y A083892 Adjacent sequences: A083889 A083890 A083891 this_sequence A083893 A083894 A083895
%Y A083892 Sequence in context: A112765 A105966 A083915 this_sequence A089814 A063100 A127475
%K A083892 nonn,base
%O A083892 1,15
%A A083892 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 08 2003

%I A089814
%S A089814 1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4,0,0,0,0,5,0,0,0,0,
%T A089814 6,0,0,0,0,9,0,0,0,0,12,0,0,0,0,13,0,0,0,0,16,0,0,0,0,21,0,0,0,0,26,0,0,
%U A089814 0,0,29,0,0,0,0,36,0,0,0,0,46,0,0,0,0,54,0,0,0,0,62,0,0,0,0,74,0
%V A089814 1,0,0,0,0,-2,0,0,0,0,1,0,0,0,0,-2,0,0,0,0,4,0,0,0,0,-4,0,0,0,0,5,0,0,0,0,-6,0,0,0,0,9,
%W A089814 0,0,0,0,-12,0,0,0,0,13,0,0,0,0,-16,0,0,0,0,21,0,0,0,0,-26,0,0,0,0,29,0,0,0,0,-36,0,0,
%X A089814 0,0,46,0,0,0,0,-54,0,0,0,0,62,0,0,0,0,-74,0
%N A089814 Expansion of Product[(1-q^(10k-5))^2,{k,Infinity}].
%H A089814 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%Y A089814 Adjacent sequences: A089811 A089812 A089813 this_sequence A089815 A089816 A089817
%Y A089814 Sequence in context: A105966 A083915 A083892 this_sequence A063100 A127475 A086014
%K A089814 sign
%O A089814 0,6
%A A089814 Eric Weisstein (eric(AT)weisstein.com), Nov 12, 2003

%I A063100
%S A063100 0,0,0,0,0,1,0,1,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
%T A063100 0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,1,0,0,0,
%U A063100 1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0
%N A063100 Compute the cototient function for the g(n) = p(n+1)-p(n)-1 composite numbers between two consecutive primes. Let the number of distinct cototient values be c[n]. Sequence gives g[n]-c[n].
%C A063100 a(n) = 0 means that all cototients in the gap are different, while 1, 2, or more means that 1 or more times inside the gap equal cototients occur.
%C A063100 Unlike totient, where phi[n+1]=phi[n] may occur (see A001274), cototients of consecutive numbers are different for n<1000000. At cases x, of A001274, cototient[x+1]=1+cototient[x].
%e A063100 Case 1: a(n) = 0; primes = {229, 233}; primes and gap = {229, 230, 231, 232, 233}; cototients = {1, 142, 111, 120, 1}, all cototients inside gap are different, thus a(n) = 0 for p(n) = p(40) = 229 prime.
%e A063100 Case 2: a(n) = 1; primes = {113, 127}; gap = {113, 114, 115, ..., 125, 126, 127}; cototients = {1, 78, 27, 60, 45, 60, 23, 88, 11, 62, 43, 64, 25, 90, 1}; seemingly 60 occurs twice, so a(n) = a(30) = g(n)-c(n) = 13-12 = 1.
%e A063100 Case 3: a(n) = 3, primes = {2861, 2879}, gap = {2861, 2862, ..., 2878, 2879}; cototients = {1, 1926, 415, 1440, 1345, 1434, 107, 1916, 169, 1910, 1191, 1440, 377, 1918, 675, 1440, 1245, 1440, 1}; observe that 1440 occurs four times, so a(n) = 3.
%Y A063100 Cf. A051953 (cototient function), A000010, A001274, A061106.
%Y A063100 Adjacent sequences: A063097 A063098 A063099 this_sequence A063101 A063102 A063103
%Y A063100 Sequence in context: A083915 A083892 A089814 this_sequence A127475 A086014 A025437
%K A063100 nonn
%O A063100 1,62
%A A063100 Labos E. (labos(AT)ana.sote.hu), Aug 07 2001

%I A127475
%S A127475 1,1,1,1,0,2,0,0,0,0,1,0,0,0,4,1,1,2,0,0,2,1,0,0,0,0,0,6,0,0,0,0,0,0,0,
%T A127475 0,0,0,0,0,0,0,0,0,0,1,1,0,0,4,0,0,0,0,4
%V A127475 1,-1,-1,-1,0,-2,0,0,0,0,-1,0,0,0,-4,1,1,2,0,0,2,-1,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,
%W A127475 0,0,0,0,0,0,1,1,0,0,4,0,0,0,0,4
%N A127475 Triangle, left column = mu(n), right border = mu(n)*phi(n), row sums = n*mu(n).
%C A127475 Left column = mu(n), A008683; right border = mu(n)*phi(n), A097945; row sums = n*mu(n), A055615: (1, -2, -3, 0, 5, 6, -7,...
%F A127475 M * A054522 as infinite lower triangular matrices, where M = mu(n) in the main diagonal with the rest zeros.
%e A127475 First few rows of the triangle are:
%e A127475 1;
%e A127475 -1, -1;
%e A127475 -1, 0, -2;
%e A127475 0, 0, 0, 0;
%e A127475 -1, 0, 0, 0, -4;
%e A127475 1, 1, 2, 0, 0, 2;
%e A127475 -1, 0, 0, 0, 0, 0, -6;
%e A127475 ...
%Y A127475 Cf. A008683, A054522, A055615, A097945, A023900.
%Y A127475 Adjacent sequences: A127472 A127473 A127474 this_sequence A127476 A127477 A127478
%Y A127475 Sequence in context: A083892 A089814 A063100 this_sequence A086014 A025437 A066032
%K A127475 tabl,uned,sign
%O A127475 1,6
%A A127475 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 15 2007

%I A086014
%S A086014 0,0,0,0,1,0,1,0,1,0,0,0,0,1,1,0,1,0,0,1,0,0,0,0,1,1,2,0,0,0,0,1,0,0,1,
%T A086014 0,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,1,
%U A086014 0,0,0,0,1,1,1,0,1,1,1,2,1,1,1,0,1,1,0,0,0,0,1,1,1,0,1,0,1,0,0,0,0,1,1
%N A086014 Number of 6's in decimal expansion of n^2.
%Y A086014 Cf. 0's A086008, 1's A086009, 2's A086010, 3's A086011, 4's A086012, 5's A086013, 7's A086015, 8's A086016, 9's A086017.
%Y A086014 Adjacent sequences: A086011 A086012 A086013 this_sequence A086015 A086016 A086017
%Y A086014 Sequence in context: A089814 A063100 A127475 this_sequence A025437 A066032 A035187
%K A086014 base,nonn
%O A086014 0,27
%A A086014 Jason Earls (jcearls(AT)cableone.net), Jul 07 2003

%I A025437
%S A025437 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,2,0,0,0,0,1,
%T A025437 0,0,1,1,0,1,1,0,0,1,2,0,0,1,2,1,0,1,2,0,1,1,0,1,0,1,3,1,0,2,2,0,0,2,2,1,
%U A025437 0,0,3,2,0,2,4,1,0,2,0,1,2,1,3,1,0,2,4,2,0,2,4,2,0,0,3,2,0,3,3,0,1,4,2,2
%N A025437 Number of partitions of n into 4 distinct squares.
%Y A025437 Adjacent sequences: A025434 A025435 A025436 this_sequence A025438 A025439 A025440
%Y A025437 Sequence in context: A063100 A127475 A086014 this_sequence A066032 A035187 A033770
%K A025437 nonn
%O A025437 0,31
%A A025437 David W. Wilson (davidwwilson(AT)comcast.net)

%I A066032
%S A066032 1,0,1,0,0,1,0,1,1,2,0,0,0,0,1,0,0,1,1,1,2,0,0,0,0,0,0,1,0,1,1,2,2,2,2,
%T A066032 3,0,0,1,1,1,1,1,1,2,0,0,0,0,1,1,1,1,1,2,0,0,0,0,0,0,0,0,0,0,1,0,0,1,2,
%U A066032 2,3,3,3,3,3,3,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,1,1,1,1,2
%N A066032 Number of ways to write n as a product with no factor larger than m (1 <= m <=n, written row by row).
%F A066032 T(1, 1) = 1. For every prime p T(p, m) = 1 if p <= m and 0 else. For composite n: T(n, m) = sum[T(n/d, d)] + I(n<=m) where the sum is over all divisors d of n except 1 and n with d <= m and I(n<=m) is 1 if n<=m and 0 else.
%e A066032 T(12, 5) = a(71) = 2, since there are 2 possibilities to write 12 as a product with no factor larger than 5 (4*3 and 3*2*2)
%p A066032 with(numtheory): T := proc(n::integer, m::integer) local i, A, summe, d: if isprime(n) then: if n <= m then RETURN(1) fi: RETURN(0): fi:
%p A066032 A := divisors(n) minus {n,1}: for d in A do: if d > m then A := A minus {d}: fi: od: summe := 0: for d in A do: summe := summe + T(n/d,d): od: if n <=m then summe := summe + 1: fi: RETURN(summe): end: A066032 := [seq(seq(T(n, m),m=1..n), n=1..16)];
%Y A066032 A001055(n) = T(n, n) is the right diagonal.
%Y A066032 Adjacent sequences: A066029 A066030 A066031 this_sequence A066033 A066034 A066035
%Y A066032 Sequence in context: A127475 A086014 A025437 this_sequence A035187 A033770 A101668
%K A066032 nonn,tabl
%O A066032 1,10
%A A066032 Ulrich Schimke (ulrschimke(AT)aol.com), Feb 11 2002

%I A035187
%S A035187 1,0,0,1,1,0,0,0,1,0,2,0,0,0,0,1,0,0,2,1,0,0,0,0,1,0,0,0,2,0,2,0,0,0,0,
%T A035187 1,0,0,0,0,2,0,0,2,1,0,0,0,1,0,0,0,0,0,2,0,0,0,2,0,2,0,0,1,0,0,0,0,0,0,
%U A035187 2,0,0,0,0,2,0,0,2,1,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,2,1,2,0,0,0,0
%N A035187 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 5.
%C A035187 Let tau be the golden ratio (1+sqrt(5))/2; let zetaQ(tau)(s)=sum(1/(Z(tau):a)^s) the Dedekind zeta function where a runs through the nonzero ideals of Z(tau) and where (Z(tau):a) is the norm of a; then zetaQ(tau)(s)=sum(n>=1,a(n)/n^s)
%H A035187 M. Baake, <a href="http://schubert.math-inf.uni-greifswald.de/preprints.html">Algebra, Combinatorics and Number Theory</a>
%H A035187 M. Baake and R. V. Moody, <a href="http://arXiv.org/abs/math.MG/9904028">Similarity submodules and root systems in four dimensions</a>, Canad. J. Math. 51 (1999), 1258-1276.
%F A035187 Sum(k=1, n, a(k)) is asymptotic to c*n where c=2*log(tau)/sqrt(5)
%F A035187 Multiplicative with a(5^e) = 1, a(p^e) = e+1 if p == 1, 4 (mod 5), a(p^e) = (1+(-1)^e)/2 if p == 2, 3 (mod 5). - Michael Somos Jun 06 2005
%F A035187 Moebius transform is period 5 sequence [1, -1, -1, 1, 0, ...]. - Michael Somos Oct 29 2005
%o A035187 (PARI) a(n)=if(n<1, 0, direuler(p=2,n,1/(1-X)/(1-kronecker(5,p)*X))[n]) /* Michael Somos Jun 06 2005 */
%o A035187 (PARI) {a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==5, 1, if((p%5==1)|(p%5==4), e+1, !(e%2))))))} /* Michael Somos Jun 06 2005 */
%o A035187 (PARI) a(n)=if(n<1, 0, sumdiv(n,d,kronecker(5,d)))
%Y A035187 Cf. A031363 (for denominators).078428.
%Y A035187 Adjacent sequences: A035184 A035185 A035186 this_sequence A035188 A035189 A035190
%Y A035187 Sequence in context: A086014 A025437 A066032 this_sequence A033770 A101668 A035202
%K A035187 nonn,mult
%O A035187 1,11
%A A035187 njas
%E A035187 Additional comments from Benoit Cloitre, Dec 29, 2002

%I A033770
%S A033770 1,1,0,1,0,0,1,0,0,0,1,1,1,0,1,1,0,1,0,0,0,2,0,0,0,0,1,
%T A033770 0,1,0,0,0,1,1,1,0,2,0,0,2,0,0,0,1,0,1,0,1,1,0,0,0,0,0
%N A033770 Product t2(q^d); d | 11, where t2 = theta2(q)/(2*q^(1/4)).
%Y A033770 Adjacent sequences: A033767 A033768 A033769 this_sequence A033771 A033772 A033773
%Y A033770 Sequence in context: A025437 A066032 A035187 this_sequence A101668 A035202 A128616
%K A033770 nonn
%O A033770 0,22
%A A033770 njas

%I A101668
%S A101668 0,1,2,0,0,0,0,1,0,1,0,1,0,1,2,0,0,1,2,0,0,1,2,0,0,1,2,0,0,0,0,1,0,1,2,
%T A101668 0,0,0,0,1,0,1,2,0,0,0,0,1,0,1,2,0,0,0,0,1,0,1,0,1,0,1,2,0,0,1,2,0,0,0,
%U A101668 0,1,0,1,0,1,0,1,2,0,0,1,2,0,0,0,0,1,0,1,0,1,0,1,2,0,0,1,2,0,0,0,0,1,0
%N A101668 Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 00.
%t A101668 Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {2, 0}, 2 -> {0, 0}})]}], {0}, 7] (from Robert G. Wilson v Feb 26 2005)
%Y A101668 Adjacent sequences: A101665 A101666 A101667 this_sequence A101669 A101670 A101671
%Y A101668 Sequence in context: A066032 A035187 A033770 this_sequence A035202 A128616 A101257
%K A101668 nonn,easy
%O A101668 0,3
%A A101668 Ralf Stephan, Dec 11 2004

%I A035202
%S A035202 1,1,0,1,1,0,0,1,1,1,2,0,0,0,0,1,0,1,2,1,0,2,0,0,1,0,0,0,2,0,2,1,0,0,0,
%T A035202 1,0,2,0,1,2,0,0,2,1,0,0,0,1,1,0,0,0,0,2,0,0,2,2,0,2,2,0,1,0,0,0,0,0,0,
%U A035202 2,1,0,0,0,2,0,0,2,1,1,2,0,0,0,0,0,2,2,1,0,0,0,0,2,0,0,1,2,1,2,0,0,0,0
%N A035202 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 20.
%C A035202 Also number of divisors of n which end in 1 or 9 minus number of divisors of n which end in 3 or 7. E.g. a(98)=2-1=1 since divisors of 98 are: 1 and 49 counting +1 each; 2, 14, and 98 counting 0 each; and 7 counting -1. - Henry Bottomley (se16(AT)btinternet.com), Jul 08 2003
%H A035202 MathNerds, <a href="http://www.mathnerds.com/best/DivisorExcess/problem.asp">An Excess of Divisors</a>.
%o A035202 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035202 Adjacent sequences: A035199 A035200 A035201 this_sequence A035203 A035204 A035205
%Y A035202 Sequence in context: A035187 A033770 A101668 this_sequence A128616 A101257 A025907
%K A035202 nonn
%O A035202 1,11
%A A035202 njas
%E A035202 More terms from Henry Bottomley (se16(AT)btinternet.com), Jul 08 2003

%I A128616
%S A128616 1,0,0,1,0,1,0,0,1,1,0,0,0,0,1,1,0,0,2,0,0,0,0,1,1,0,0,0,0,0,2,0,0,2,0,
%T A128616 1,0,0,0,1,0,0,0,0,0,2,0,0,1,0,2,0,0,1,0,0,0,0,0,1,2,0,0,1,0,0,0,0,2,0,
%U A128616 0,0,0,0,0,2,0,0,2,0,1,0,0,0,2,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,0,0,0,0,0
%N A128616 Expansion of q* psi(q^3)* psi(q^5) in powers of q where psi() is a Ramanujan theta function.
%D A128616 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(iv).
%F A128616 Expansion of (eta(q^6)* eta(q^10))^2/ (eta(q^3)* eta(q^5)) in powers of q.
%F A128616 Euler transform of period 30 sequence [ 0, 0, 1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 1, 0, 0, -1, 1, 0, 1, 0, 0, -2, ...].
%F A128616 For n>0, n in A028957 equivalent to a(n) nonzero. If a(n) nonzero, a(n)= A082451(n), and a(n)= A121362(n).
%F A128616 a(n)= (A082451(n)+ A121362(n))/2.
%F A128616 G.f.: x* Product_{k>0} (1-x^(3k))* (1-x^(5k))* (1+x^(6k))^2* (1+x^(10k))^2.
%o A128616 (PARI) {a(n)= if(n<1, 0, sumdiv(n, d, kronecker(-60, d) +kronecker(20, d)* kronecker(-3, n/d) )/2)}
%o A128616 (PARI) {a(n)= local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x^6+A)* eta(x^10+A))^2/ (eta(x^3+A)* eta(x^5+A)), n))}
%Y A128616 Adjacent sequences: A128613 A128614 A128615 this_sequence A128617 A128618 A128619
%Y A128616 Sequence in context: A033770 A101668 A035202 this_sequence A101257 A025907 A024157
%K A128616 nonn
%O A128616 1,19
%A A128616 Michael Somos, Mar 13 2007

%I A101257
%S A101257 0,0,0,0,0,1,0,0,0,1,0,1,0,1,2,0,0,0,0,1,1,1,0,2,0,1,0,3,0,1,0,0,2,1,2,
%T A101257 0,0,1,1,3,0,1,0,3,4,1,0,2,0,0,2,1,0,3,1,1,1,1,0,4,0,1,2,0,3,5,0,1,2,3,
%U A101257 0,1,0,1,0,3,4,1,0,2,0,1,0,5,2,1,2,3,0,1,6,3,1,1,4,4,0,0,2,0,0,5,0,5,1
%N A101257 Remainder when the least divisor of n greater than the square root of n (A033677(n)) is divided by the greatest divisor of n less than the square root of n (A033676(n)).
%C A101257 Given n points, sort them into the most-square rectangular point lattice possible. Now sort the points into square point lattices of dimension equal to the lesser dimension of the earlier rectangle. a(n) is the number of points left over. a(n) is trivially 0 for prime numbers n (the most-square and only rectangular point lattice on a prime number of points is a linear point lattice). a(n) != 0 iff n is a member of A080363
%H A101257 Eric Weisstein et al., <a href="http://mathworld.wolfram.com/PointLattice.html">"Point Lattice."</a>
%H A101257 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Divisor.html">"Divisor."</a>
%e A101257 a(6)=1 because the least divisor of 6 greater than sqrt(6) is 3, the greater divisor of 6 less than sqrt(6) is 2, and 3 mod 2 = 1
%t A101257 num[n_] := If[OddQ[DivisorSigma[0, n]], Sqrt[n], Divisors[n][[DivisorSigma[0, n]/2 + 1]]] den[n_] := If[OddQ[DivisorSigma[0, n]], Sqrt[n], Divisors[n][[DivisorSigma[0, n]/2]]] Table[Mod[num[n], den[n]], {n, 1, 128}]
%Y A101257 Cf. A033676, A033677, A080363.
%Y A101257 Adjacent sequences: A101254 A101255 A101256 this_sequence A101258 A101259 A101260
%Y A101257 Sequence in context: A101668 A035202 A128616 this_sequence A025907 A024157 A039968
%K A101257 base,nonn
%O A101257 1,15
%A A101257 Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 17 2004

%I A025907
%S A025907 1,0,0,0,0,0,1,0,0,0,0,1,2,0,0,0,0,1,2,0,0,0,1,2,3,0,0,
%T A025907 0,1,2,3,0,0,1,2,3,4,0,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,
%U A025907 5,1,2,3,4,5,6,1,2,3,4,5,7,2,3,4,5,6,8,2,3,4,5,7,9,3
%N A025907 Expansion of 1/((1-x^6)(1-x^11)(1-x^12)).
%Y A025907 Adjacent sequences: A025904 A025905 A025906 this_sequence A025908 A025909 A025910
%Y A025907 Sequence in context: A035202 A128616 A101257 this_sequence A024157 A039968 A092037
%K A025907 nonn
%O A025907 0,13
%A A025907 njas

%I A024157
%S A024157 0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,1,2,0,0,2,0,0,1,2,0,4,0,1,4,0,4,2,0,
%T A024157 0,4,5,0,6,0,3,6,0,0,9,0,0,6,3,0,3,6,7,6,0,0,14,0,0,9,4,8,10,0,4,8,12,0,9,0,
%U A024157 0,9,5,10,12,0,17,11,0,0,21,10,0,10,10,0,18,12,6,10,0,12,22,0,1,13,0
%N A024157 Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n such that n divides abc.
%Y A024157 Adjacent sequences: A024154 A024155 A024156 this_sequence A024158 A024159 A024160
%Y A024157 Sequence in context: A128616 A101257 A025907 this_sequence A039968 A092037 A037134
%K A024157 nonn
%O A024157 1,15
%A A024157 Clark Kimberling (ck6(AT)evansville.edu)

%I A039968
%S A039968 1,0,1,2,0,0,0,0,1,2,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,0,2,1,0,0,0,0,
%T A039968 2,1,0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A039968 0,0,0,0,0,0,0,0,0,0,1,2,0,2,1,0,0,0,0,2,1,0,1,2,0,0,0,0,0,0,0,0,0,0,0
%N A039968 An example of a d-perfect sequence.
%H A039968 D. Kohel, S. Ling and C. Xing, <a href="http://magma.maths.usyd.edu.au/users/kohel/documents/perfect.ps">Explicit Sequence Expansions</a>
%H A039968 D. Kohel, S. Ling and C. Xing, <a href="http://www.maths.usyd.edu.au/u/kohel/doc/perfect.ps">Explicit Sequence Expansions</a>
%F A039968 a(n) = ((-1)^(n+1)*A005043(n-1)) mod 3 - Christian G. Bower (bowerc(AT)usa.net), Jun 12 2005 - Christian G. Bower (bowerc(AT)usa.net), Jun 12 2005
%Y A039968 Adjacent sequences: A039965 A039966 A039967 this_sequence A039969 A039970 A039971
%Y A039968 Sequence in context: A101257 A025907 A024157 this_sequence A092037 A037134 A001343
%K A039968 nonn
%O A039968 1,4
%A A039968 njas
%E A039968 More terms from Christian G. Bower (bowerc(AT)usa.net), Jun 12 2005

%I A092037
%S A092037 1,1,0,1,2,0,0,0,0,1,2,0,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,0,2,2,0,0,0,
%T A092037 0,2,2,0,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A092037 0,0,0,0,0,0,0,0,0,0,0,1,2,0,2,2,0,0,0,0,2,2,0,2,2,0,0,0,0,0,0,0,0,0,0
%N A092037 A092255 mod 3.
%F A092037 a(n)=0 iff n is in A074940; a(n)=1 iff n is a power of 3; a(n)=2 iff n is in A092428
%Y A092037 Adjacent sequences: A092034 A092035 A092036 this_sequence A092038 A092039 A092040
%Y A092037 Sequence in context: A025907 A024157 A039968 this_sequence A037134 A001343 A022882
%K A092037 nonn
%O A092037 0,5
%A A092037 Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 27 2004

%I A037134
%S A037134 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,1,0,
%T A037134 0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,2,0,0,0,0,2,0,0,0,0,1,1,0,1,0,
%U A037134 0,0,0,0,2,0,1,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,1,0,1
%N A037134 Number of ways of writing n as a+b where a is abundant (or perfect), b is deficient (or perfect) and s(a)-a=b-s(b) (s(x)=sigma(x)-x).
%H A037134 Naohiro Nomoto, <a href="http://www.geocities.co.jp/Technopolis/1793/Tao-e.htm">Sequence of Yin and Yang</a>
%e A037134 E.g. 17=5+12, 5-s(5)=s(12)-12, a(17)=1; 34=6+28, a(34)=1.
%Y A037134 Adjacent sequences: A037131 A037132 A037133 this_sequence A037135 A037136 A037137
%Y A037134 Sequence in context: A024157 A039968 A092037 this_sequence A001343 A022882 A000089
%K A037134 nonn
%O A037134 1,56
%A A037134 Naohiro Nomoto (6284968128(AT)geocities.co.jp)

%I A001343
%S A001343 1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4,0,
%T A001343 0,0,0,6,0,0,0,0,6,0,0,0,0,9,0,0,0,0,9,0,0,0,0,13,0,0,0,
%U A001343 0,13,0,0,0,0,18,0,0,0,0,18,0,0,0,0,24,0,0,0,0,24,0,0,0
%N A001343 Number of (unordered) ways of making change for n cents using coins of 5, 10, 20, 50, 100 cents.
%D A001343 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A001343 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H A001343 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=185">Encyclopedia of Combinatorial Structures 185</a>
%H A001343 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mag.html#change">Index entries for sequences related to making change.</a>
%p A001343 1/(1-x^5)/(1-x^10)/(1-x^20)/(1-x^50)/(1-x^100)
%t A001343 a[n_] := SeriesTerm[1/((1 - x^5)(1 - x^10)(1 - x^20)(1 - x^50)(1 - x^100)), {x, 0, n}]
%Y A001343 Adjacent sequences: A001340 A001341 A001342 this_sequence A001344 A001345 A001346
%Y A001343 Sequence in context: A039968 A092037 A037134 this_sequence A022882 A000089 A051907
%K A001343 nonn
%O A001343 0,11
%A A001343 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)

%I A022882
%S A022882 0,0,0,1,0,0,0,2,0,0,0,0,2,0,0,0,3,0,0,0,0,1,0,1,0,3,0,0,1,0,1,0,1,
%T A022882 0,3,0,0,1,0,2,0,0,0,3,0,0,0,0,3,0,0,0,5,0,0,0,0,3,0,0,0,2,0,1,0,0,
%U A022882 3,0,1,0,3,0,2,0,0,2,0,1,0,3,0,0,0,2,0,0,1,0,5,0
%N A022882 The number of numbers [ [ ix ]jx ] that equal n, where i >= 1, j >= 1 and x=sqrt(5).
%Y A022882 Adjacent sequences: A022879 A022880 A022881 this_sequence A022883 A022884 A022885
%Y A022882 Sequence in context: A092037 A037134 A001343 this_sequence A000089 A051907 A093569
%K A022882 nonn
%O A022882 1,8
%A A022882 Clark Kimberling (ck6(AT)evansville.edu)

%I A000089
%S A000089 1,1,0,0,2,0,0,0,0,2,0,0,2,0,0,0,2,0,0,0,0,0,0,0,2,2,0,0,2,0,0,0,0,2,0,
%T A000089 0,2,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,2,0,0,2,0,0,0,4,0,0,0,0,0,
%U A000089 0,0,2,2,0,0,0,0,0,0,0,2,0,0,4,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0
%N A000089 Number of solutions to x^2 + 1 == 0 (mod n).
%C A000089 Number of elliptic points of order 2 for GAMMA_0 (n).
%D A000089 M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
%D A000089 Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations. J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
%D A000089 G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2).
%H A000089 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000089.txt">Table of n, a(n) for n=1..2000</a>
%H A000089 M. Baake and U. Grimm, <a href="http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-392">Quasicrystalline combinatorics</a>
%H A000089 S. R. Finch and Pascal Sebah, <a href="http://arXiv.org/abs/math.NT/0604465">Squares and Cubes Modulo n</a> (arXiv: math.NT/0604465).
%F A000089 a(n) = 0 if 4|n, else a(n) = Product_{ p | N } (1 + Legendre(-1, p) ), where we use the definition that Legendre(-1, 2) = 0, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4. This is Shimura's definition, which is different from Maple's.
%F A000089 Dirichlet series: (1+2^(-s))*Product (1+p^(-s))/(1-p^(-s)) (p=1 mod 4).
%F A000089 Multiplicative with a(p^e) = 1 if p = 2 and e = 1; 0 if p = 2 and e > 1; 2 if p == 1 (mod 4); 0 if p == 3 (mod 4). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
%p A000089 with(numtheory); A000089 := proc (n) local i, s; if modp(n,4) = 0 then RETURN(0) fi; s := 1; for i in divisors(n) do if isprime(i) and i > 2 then s := s*(1+eval(legendre(-1,i))) fi od; s end: (Gene Smith, May 22 2006)
%t A000089 Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 3 ]==2, 0, Count[ Array[ Mod[ #^2+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
%Y A000089 Adjacent sequences: A000086 A000087 A000088 this_sequence A000090 A000091 A000092
%Y A000089 Sequence in context: A037134 A001343 A022882 this_sequence A051907 A093569 A073091
%K A000089 nonn,nice,mult
%O A000089 1,5
%A A000089 njas

%I A051907
%S A051907 1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,
%T A051907 0,1,1,0,0,0,0,1,0,2,0,0,0,0,2,0,1,1,1,1,0,2,0,1,1,1,2,0,4,1,3,4,0,2,0,
%U A051907 6,0,1,2,1,3,0,4,2,1,5,5,3,2,3,3,5,5,5,2,1,12,5,4,11,4,5,2,11,3,5
%N A051907 Number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.
%H A051907 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ed.html#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A051907 1 = 1/2+1/4+1/9+1/12+1/18 = 1/2+1/5+1/6+1/12+1/20. The sum of the denominators of each of these is 45, these are the only 2 with sum of denominators = 45, so a(45)=2.
%Y A051907 A051882 lists n such that a(n)=0.
%Y A051907 Adjacent sequences: A051904 A051905 A051906 this_sequence A051908 A051909 A051910
%Y A051907 Sequence in context: A001343 A022882 A000089 this_sequence A093569 A073091 A125250
%K A051907 nonn
%O A051907 1,45
%A A051907 Jud McCranie (j.mccranie(AT)comcast.net), Dec 16 1999
%E A051907 R. L. Graham showed that a(n)>0 for n>77.

%I A093569
%S A093569 0,0,0,0,2,0,0,0,0,2,0,2,0,2,0,2,0,2,0,0,0,0,0,0,2,0,0,0,6,0,0,0,2,0,0,
%T A093569 0,0,0,0,2,0,0,0,0,0,2,0,0,4,0,0,0,0,0,2,0,2,2,0,0,0,0,0,0,2,0,0,0,2,0,
%U A093569 2,0,0,0,2,0,0,2,2,2,0,2,0,2,2,0,0,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0
%N A093569 For p = prime(n), the number of integers k < p-1 such that p divides A001008(k), the numerator of the harmonic number H(k).
%C A093569 It is well-known that prime p >= 3 divides the numerator of H(p-1). For primes p in A092194, there are integers k < p-1 for which p divides the numerator of H(k). Interestingly, if p divides A001008(k) for k < p-1, then p divides A001008(p-k-1). Hence the terms of this sequence are usually even. The only exceptions are the two known Wieferich primes 1093 and 3511, A001220, which have 3 values of k < p-1 for which p divides A001008(k), one being k = (p-1)/2.
%H A093569 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H A093569 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WieferichPrime.html">Wieferich Prime</a>
%e A093569 a(5) = 2 because 11 = prime(5) and there are 2 values, k = 3 and 7, such that 11 divides A001008(k).
%t A093569 len=500; Table[p=Prime[i]; cnt=0; k=1; While[k<p-1, If[Mod[Numerator[HarmonicNumber[k]], p]==0, cnt++ ]; k++ ]; cnt, {i, len}]
%Y A093569 Cf. A001008, A001220, A092194.
%Y A093569 Adjacent sequences: A093566 A093567 A093568 this_sequence A093570 A093571 A093572
%Y A093569 Sequence in context: A022882 A000089 A051907 this_sequence A073091 A125250 A048113
%K A093569 nonn
%O A093569 1,5
%A A093569 T. D. Noe (noe(AT)sspectra.com), Apr 01 2004

%I A073091
%S A073091 1,0,0,1,0,1,0,0,1,1,2,0,0,0,0,2,1,1,1,0,0,0,0,1,2,0,1,0,0,0,2,0,1,2,0,
%T A073091 1,0,1,0,2,0,0,1,2,1,1,0,0,3,0,2,0,0,1,0,0,2,0,0,1,0,0,0,3,0,1,2,1,2,0,
%U A073091 1,1,0,0,0,3,0,0,2,0,2,1,1,0,2,2,0,0,1,1,0,0,0,2,0,1,0,0,3,3
%N A073091 Number of ways to express n as (x^3+y^3+z^3)/(x+y+z) 1<=x<=y<=z<=n.
%o A073091 (PARI) a(n)=sum(x=1,n,sum(y=1,x,sum(z=1,y,if((x^3+y^3+z^3)/(x+y+z)-n,0,1))))
%Y A073091 Adjacent sequences: A073088 A073089 A073090 this_sequence A073092 A073093 A073094
%Y A073091 Sequence in context: A000089 A051907 A093569 this_sequence A125250 A048113 A028961
%K A073091 easy,nonn
%O A073091 1,11
%A A073091 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 18 2002

%I A125250
%S A125250 1,0,0,0,1,0,0,1,1,0,0,0,2,0,0,0,0,2,2,0,0,0,0,1,5,1,0,0,0,0,0,5,5,0,0,
%T A125250 0,0,0,0,3,11,3,0,0,0,0,0,0,1,13,13,1,0,0,0,0,0,0,0,9,26,9,0,0,0,0,0,0,
%U A125250 0,0,4,32,32,4,0,0,0,0,0,0,0,0,1,26,63,26,1,0,0,0,0,0,0,0,0,0,14,80,80
%N A125250 Array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n<1 or k<1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).
%C A125250 It appears that the main diagonal (1,1,2,5,11,...) is A05186 (Whitney number of level n of the lattice of the ideals of the fence of size 2 n) that the diagonals (0,1,2,5,13,...) adjacent to the main diagonal are A110320 (Number of blocks in all RNA secondary structures with n nodes), and that the n-th antidiagonal sum = A094686(n-1) (a Fibonacci convolution). The n-th row sum = A002605(n).
%F A125250 A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n<1 or k<1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1)
%e A125250 Example:
%e A125250 Array starts as
%e A125250 1 0 0 0 0 0 0 ...
%e A125250 0 1 1 0 0 0 0 ...
%e A125250 0 1 2 2 1 0 0 ...
%e A125250 ...
%o A125250 (PARI) A=matrix(22,22);A[1,1]=1;A[2,2]=1;A[2,1]=0;A[1,2]=0;A[3,2]=1;A[2,3]=1; for(n=3,22,for(k=3,22,A[n,k]=A[n-2,k-2]+A[n-1,k-2]+A[n-2,k-1]+A[n-1,k-1])); for(n=1,22,for(i=1,n,print1(A[n-i+1,i],", ")))
%Y A125250 Cf. A051286, A110320, A002605.
%Y A125250 Adjacent sequences: A125247 A125248 A125249 this_sequence A125251 A125252 A125253
%Y A125250 Sequence in context: A051907 A093569 A073091 this_sequence A048113 A028961 A110177
%K A125250 nonn,tabl
%O A125250 1,13
%A A125250 Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Jan 15 2007

%I A048113
%S A048113 0,0,0,0,1,0,0,1,1,0,0,0,2,0,0,0,0,2,2,0,0,0,0,2,4,2,0,0,0,0,0,6,6,0,0,
%T A048113 0,0,0,0,6,12,6,0,0,0,0,0,0,6,18,18,6,0,0,0,0,0,0,0,24,36,24,0,0,0,0,0,
%U A048113 0,0,0,24,60,60,24,0,0,0,0,0,0,0,0,24,84
%N A048113 Triangular array T read by rows: T(h,k)=number of paths consisting of steps from (1,1) to (h,k) such that each step has length 1 directed up or right and each vertex (i,j) satisfies i/2<=j<=2i, for h=0,1,2,... and k=0,1,2,...
%e A048113 Rows: {0}; {0,0}; {0,1,0}; {0,1,1,0}; ...
%Y A048113 Adjacent sequences: A048110 A048111 A048112 this_sequence A048114 A048115 A048116
%Y A048113 Sequence in context: A093569 A073091 A125250 this_sequence A028961 A110177 A036273
%K A048113 nonn,tabl
%O A048113 1,13
%A A048113 Clark Kimberling (ck6(AT)evansville.edu)

%I A028961
%S A028961 1,0,0,2,0,0,0,0,2,2,0,0,2,2,0,0,2,0,0,0,0,0,0,0,2,0,0,2,0,2,0,2,2,
%T A028961 0,0,0,2,0,0,2,0,2,0,0,0,0,0,2,4,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,
%U A028961 0,0,0,2,0,2,4,2,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,6,0,0
%N A028961 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 8 ].
%H A028961 John Cannon, <a href="http://www.research.att.com/~njas/sequences/b028961.txt">Table of n, a(n) for n = 0..10000</a>
%Y A028961 Adjacent sequences: A028958 A028959 A028960 this_sequence A028962 A028963 A028964
%Y A028961 Sequence in context: A073091 A125250 A048113 this_sequence A110177 A036273 A112166
%K A028961 nonn
%O A028961 0,4
%A A028961 njas

%I A110177
%S A110177 0,0,2,0,0,0,0,2,2,2,0,0,0,0,2,0,0,0,0,2,2,0,0,0,0,0,0,0,0,2,0,4,2,0,0,
%T A110177 0,0,0,2,2,0,0,0,0,0,0,0,0,2,0,2,0,0,0,4,2,4,0,0,0,0,2,4,0,0,0,0,0,4,2,
%U A110177 0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,4,2,0,2,0,2,2,2,0,2,0,0,2,0,0,0,0,2,4
%N A110177 Number of solutions 0<k<n to the equation sigma(n)=sigma(k)+sigma(n-k), where sigma is the sum of divisors function.
%C A110177 The number of solutions is always even because k=n/2 cannot be a solution for even n.
%t A110177 a[n_] := Select[Range[n-1], DivisorSigma[1, n]==DivisorSigma[1, n-# ]+DivisorSigma[1, # ]&]; Table[Length[a[n]], {n, 150}]
%Y A110177 Cf. A110176 (least k such that sigma(n)=sigma(k)+sigma(n-k)).
%Y A110177 Adjacent sequences: A110174 A110175 A110176 this_sequence A110178 A110179 A110180
%Y A110177 Sequence in context: A125250 A048113 A028961 this_sequence A036273 A112166 A112167
%K A110177 nonn
%O A110177 1,3
%A A110177 T. D. Noe (noe(AT)sspectra.com), Jul 15 2005

%I A036273
%S A036273 1,2,0,0,0,0,2,2,2,2,0,0,2,2,4,2,2,2,0,2,0,2,0,2,0,0,8,8,8,6,2,
%T A036273 4,4,0,2,2,2,2,0,4,6,6,8,2,2,2,6,0,2,2,0,0,4,4,4,2,2,6,2,2,2,8,
%U A036273 8,6,2,0,2,6,0,0,2,0,2,0,2,2,2,4,6,0,4,2,2,0,2,0,2,6,2,4,0,0,2
%N A036273 Absolute values of differences of A036272.
%Y A036273 Cf. A036262, A036263.
%Y A036273 Adjacent sequences: A036270 A036271 A036272 this_sequence A036274 A036275 A036276
%Y A036273 Sequence in context: A048113 A028961 A110177 this_sequence A112166 A112167 A037213
%K A036273 nonn
%O A036273 0,2
%A A036273 njas

%I A112166
%S A112166 1,2,0,0,0,0,2,4,0,0,0,0,1,6,0,0,0,0,2,12,0,0,0,0,4,18,0,0,0,0,4,28,0,0,
%T A112166 0,0,5,44,0,0,0,0,6,64,0,0,0,0,9,92,0,0,0,0,12,132,0,0,0,0,13,186,0,0,0,
%U A112166 0,16,256,0,0,0,0,21,352,0,0,0,0,26,476,0,0,0,0,29,638,0,0,0,0,36
%V A112166 1,2,0,0,0,0,-2,4,0,0,0,0,1,6,0,0,0,0,-2,12,0,0,0,0,4,18,0,0,0,0,-4,28,0,0,0,0,5,44,0,
%W A112166 0,0,0,-6,64,0,0,0,0,9,92,0,0,0,0,-12,132,0,0,0,0,13,186,0,0,0,0,-16,256,0,0,0,0,21,
%X A112166 352,0,0,0,0,-26,476,0,0,0,0,29,638,0,0,0,0,-36
%N A112166 McKay-Thompson series of class 24i for the Monster group.
%D A112166 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A112166 T24i = 1/q +2*q -2*q^11 +4*q^13 +q^23 +6*q^25 -2*q^35 +12*q^37 +...
%Y A112166 Adjacent sequences: A112163 A112164 A112165 this_sequence A112167 A112168 A112169
%Y A112166 Sequence in context: A028961 A110177 A036273 this_sequence A112167 A037213 A092197
%K A112166 sign
%O A112166 0,2
%A A112166 Michael Somos, Aug 28 2005

%I A112167
%S A112167 1,2,0,0,0,0,2,4,0,0,0,0,1,6,0,0,0,0,2,12,0,0,0,0,4,18,0,0,0,0,4,28,0,0,
%T A112167 0,0,5,44,0,0,0,0,6,64,0,0,0,0,9,92,0,0,0,0,12,132,0,0,0,0,13,186,0,0,0,
%U A112167 0,16,256,0,0,0,0,21,352,0,0,0,0,26,476,0,0,0,0,29,638,0,0,0,0,36
%V A112167 1,-2,0,0,0,0,-2,-4,0,0,0,0,1,-6,0,0,0,0,-2,-12,0,0,0,0,4,-18,0,0,0,0,-4,-28,0,0,0,0,5,
%W A112167 -44,0,0,0,0,-6,-64,0,0,0,0,9,-92,0,0,0,0,-12,-132,0,0,0,0,13,-186,0,0,0,0,-16,-256,0,
%X A112167 0,0,0,21,-352,0,0,0,0,-26,-476,0,0,0,0,29,-638,0,0,0,0,-36
%N A112167 McKay-Thompson series of class 24j for the Monster group.
%D A112167 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A112167 T24j = 1/q -2*q -2*q^11 -4*q^13 +q^23 -6*q^25 -2*q^35 -12*q^37 +...
%Y A112167 Adjacent sequences: A112164 A112165 A112166 this_sequence A112168 A112169 A112170
%Y A112167 Sequence in context: A110177 A036273 A112166 this_sequence A037213 A092197 A083804
%K A112167 sign
%O A112167 0,2
%A A112167 Michael Somos, Aug 28 2005

%I A037213
%S A037213 0,1,0,0,2,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,
%T A037213 0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,
%U A037213 0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A037213 Expansion of sum(n*q^(n^2), n=0..inf).
%C A037213 Multiplicative with a(p^(2e)) = p^e, a(p^(2e+1)) = 0. Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) Jun 09, 2005.
%F A037213 Dirichlet generating function: zeta(2s-1). - Franklin T. Adams-Watters, Sep 11 2005.
%Y A037213 Adjacent sequences: A037210 A037211 A037212 this_sequence A037214 A037215 A037216
%Y A037213 Sequence in context: A036273 A112166 A112167 this_sequence A092197 A083804 A028597
%K A037213 nonn,mult
%O A037213 0,5
%A A037213 njas

%I A092197
%S A092197 0,0,0,2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2,0,
%T A092197 0,0,0,3,2,2,2,2,4,2,2,2,2,5,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3,0,
%U A092197 0,0,0,2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3,3,3,3,3,5,3,3,3,3,6,0,0,0,0,2,0
%N A092197 Brevity advantage of "new style" over "old style" Roman numerals.
%F A092197 Equals A092196 - A006968.
%e A092197 a(4)=2 because old style takes four letters (IIII) versus new style's two (IV).
%Y A092197 Cf. A006968 A092196.
%Y A092197 Adjacent sequences: A092194 A092195 A092196 this_sequence A092198 A092199 A092200
%Y A092197 Sequence in context: A112166 A112167 A037213 this_sequence A083804 A028597 A028617
%K A092197 base,easy,nonn
%O A092197 1,4
%A A092197 Marc LeBrun (mlb(AT)well.com), Feb 24 2004

%I A083804
%S A083804 1,2,0,0,0,0,3,2,1,10,0,8,16,8,15,20,9,4,24,20,13,28,25,28,25,30,32,20,
%T A083804 9,10,27,2,16,10,5,36,26,10,33,20,33,46,26,28,30,50,4,48,13,50,3,12,12,
%U A083804 6,30,64,53,56,61,60,69,36,12,8,65,74,66,4,44,20,3,76,31,62,75,6,85,56
%N A083804 a(1) = 1. If the (n-1)th partial concatenation is divisible by n then a(n) = 0. Otherwise, a(n) is the smallest positive number such that the n-th partial concatenation is divisible by n.
%e A083804 1,12,120,1200,12000,120000 1200003 etc. are multiples of 1,2,3,4,5,6 and 7 respectively.
%Y A083804 Cf. A083805.
%Y A083804 Cf. A051883, A073894, A099552.
%Y A083804 Adjacent sequences: A083801 A083802 A083803 this_sequence A083805 A083806 A083807
%Y A083804 Sequence in context: A112167 A037213 A092197 this_sequence A028597 A028617 A006792
%K A083804 base,easy,nonn
%O A083804 1,2
%A A083804 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 08 2003
%E A083804 Corrected and extended by David Wasserman (wasserma(AT)spawar.navy.mil), Nov 30 2004

%I A028597
%S A028597 1,0,0,0,2,0,0,0,0,4,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,
%T A028597 0,0,0,0,0,2,4,0,0,6,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,
%U A028597 0,0,0,4,0,0,0,0,0,0,2,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0
%N A028597 Expansion of (theta_3(z)*theta_3(8z)+theta_2(z)*theta_2(8z)).
%Y A028597 Adjacent sequences: A028594 A028595 A028596 this_sequence A028598 A028599 A028600
%Y A028597 Sequence in context: A037213 A092197 A083804 this_sequence A028617 A006792 A011992
%K A028597 nonn
%O A028597 0,5
%A A028597 njas

%I A028617
%S A028617 1,0,2,0,0,0,0,4,2,0,0,4,0,0,0,0,0,0,2,4,0,0,0,0,0,0,2,
%T A028617 0,4,0,0,4,2,0,4,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,2,0,0,0,
%U A028617 0,0,0,0,4,4,0,0,0,4,0,0,0,4,0,0,0,4,2,0,0,0,4,0,0,0,0
%N A028617 Expansion of (theta_3(z)*theta_3(13z)+theta_2(z)*theta_2(13z)).
%Y A028617 Adjacent sequences: A028614 A028615 A028616 this_sequence A028618 A028619 A028620
%Y A028617 Sequence in context: A092197 A083804 A028597 this_sequence A006792 A011992 A107503
%K A028617 nonn
%O A028617 0,3
%A A028617 njas

%I A006792 M0009
%S A006792 2,0,0,0,0,4,8,0,4,0,82,0,0,0,112,0,132,0,66,0,1124,0
%N A006792 Number of n-node vertex-transitive graphs which are not Cayley graphs.
%D A006792 McKay, Brendan D.; Praeger, Cheryl E.; Vertex-transitive graphs which are not Cayley graphs. I. J. Austral. Math. Soc. Ser. A 56 (1994), no. 1, 53-63.
%D A006792 McKay, Brendan D.; Royle, Gordon F.; The transitive graphs with at most 26 vertices. Ars Combin. 30 (1990), 161-176.
%H A006792 Steven Skiena, <a href="http://www.cs.sunysb.edu/~skiena/combinatorica/graphs/">A Database of Graphs in Combinatorica Format</a>.
%Y A006792 Adjacent sequences: A006789 A006790 A006791 this_sequence A006793 A006794 A006795
%Y A006792 Sequence in context: A083804 A028597 A028617 this_sequence A011992 A107503 A135468
%K A006792 nonn
%O A006792 10,1
%A A006792 njas
%E A006792 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 30 2007

%I A011992
%S A011992 2,0,0,0,0,4,9,0,9,71,0,0,201,0,175,0,0,2939,19543,0,10841,609483
%N A011992 (n,3,2) difference families over Z_n.
%D A011992 CRC Handbook of Combinatorial Designs, 1996, p. 285.
%Y A011992 Adjacent sequences: A011989 A011990 A011991 this_sequence A011993 A011994 A011995
%Y A011992 Sequence in context: A028597 A028617 A006792 this_sequence A107503 A135468 A003196
%K A011992 nonn,hard
%O A011992 7,1
%A A011992 njas

%I A107503
%S A107503 1,0,0,0,2,0,0,0,0,6,4,0,10,2,6,0,8,6
%N A107503 Coefficients of a certain theta series.
%C A107503 See reference for details.
%D A107503 W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.
%Y A107503 Adjacent sequences: A107500 A107501 A107502 this_sequence A107504 A107505 A107506
%Y A107503 Sequence in context: A028617 A006792 A011992 this_sequence A135468 A003196 A062977
%K A107503 nonn
%O A107503 0,5
%A A107503 njas, May 28 2005

%I A135468
%S A135468 0,0,1,0,0,0,0,2,0,0,0,0,18,0,0,0,0,36,0,0,0,0,122,0,0,0,0,144,0,0,0,0,
%T A135468 852,0,0,0,0,96,0,0,0,0,5244,0,0,0,0,4538,0,0,0,0,7332,0,0,0,0,151164,0,
%U A135468 0,0,0,67896,0,0,0,0,236208,0,0,0,0,1405224,0,0,0,0,577248,0,0,0,0,20031881
%V A135468 0,0,1,0,0,0,0,-2,0,0,0,0,-18,0,0,0,0,36,0,0,0,0,122,0,0,0,0,-144,0,0,0,0,
%W A135468 -852,0,0,0,0,-96,0,0,0,0,5244,0,0,0,0,-4538,0,0,0,0,-7332,0,0,0,0,151164,0,
%X A135468 0,0,0,67896,0,0,0,0,236208,0,0,0,0,-1405224,0,0,0,0,577248,0,0,0,0,-20031881
%N A135468 Numerators of expansion of eta(5t)^2*eta(10t)^(13/5)*eta(20t)^(-9/5)*eta(40t)^(6/5).
%D A135468 Ishikawa, T., Congruences between binomial coefficients binom(2f,f) and Fourier coefficients of certain eta-products, Hiroshima Math. J. 22 (1992), no. 3, 583-590.
%e A135468 q^2-2*q^7-18/5*q^12+36/5*q^17+122/25*q^22-144/25*q^27-...
%Y A135468 Adjacent sequences: A135465 A135466 A135467 this_sequence A135469 A135470 A135471
%Y A135468 Sequence in context: A006792 A011992 A107503 this_sequence A003196 A062977 A072325
%K A135468 sign,frac
%O A135468 2,8
%A A135468 njas, Feb 07 2008

%I A003196 M0010
%S A003196 1,0,0,0,0,0,2,0,0,0,0,24,26,0,0,48,252,720,438,192,984,1008,12924,19536,3062,
%T A003196 8280,26694,153536,507948,406056,79532,729912,631608,9279376,15771600,
%U A003196 7467336,10935114,21835524,112752684,400576168,410287368
%V A003196 1,0,0,0,0,0,-2,0,0,0,0,-24,26,0,0,-48,-252,720,-438,-192,-984,-1008,12924,-19536,3062,
%W A003196 -8280,26694,153536,-507948,406056,-79532,729912,631608,-9279376,15771600,
%X A003196 -7467336,10935114,-21835524,-112752684,400576168,-410287368
%N A003196 Magnetization series for face-centered cubic lattice.
%D A003196 C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421.
%D A003196 M. F. Sykes, J. W. Essam and D. S. Gaunt, Derivation of low-temperature expansions for the Ising model of a ferromagnet and an antiferromagnet, J. Math. Phys. 6 (1965), 283-298.
%D A003196 M. F. Sykes et al., Derivation of low-temperature expansions for Ising model VI. Three-dimensional lattices - temperature grouping, J. Phys. A 6 (1973), 1507-1516.
%Y A003196 Adjacent sequences: A003193 A003194 A003195 this_sequence A003197 A003198 A003199
%Y A003196 Sequence in context: A011992 A107503 A135468 this_sequence A062977 A072325 A076948
%K A003196 sign,nice
%O A003196 0,7
%A A003196 njas

%I A062977
%S A062977 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,1,0,0,0,0,0,0,0,
%T A062977 0,0,0,0,2,0,0,0,1,1,0,0,3,0,1,0,1,0,2,0,2,0,0,0,1,0,0,1,0,0,0,0,1,0,0,
%U A062977 0,1,0,0,1,1,0,0,0,3,0,0,0,1,0,0,0,2,0,1,0,1,0,0,0,4,0,1,1,0,0,0,0,2,0
%N A062977 Difference between largest and smallest positive exponent in prime factorization of n.
%F A062977 a(n) =A051903(n)-A051904(n)
%e A062977 a(24)=2 since 120=2^3*3^1 and max(3,1)-min(3,1)=3-1=2; a(25)=0 since 25=5^2 and max(2)-min(2)=2-2=0.
%Y A062977 If a(n) is positive then n is in A059404, while if a(n) is 0 then n is in A062770.
%Y A062977 Adjacent sequences: A062974 A062975 A062976 this_sequence A062978 A062979 A062980
%Y A062977 Sequence in context: A107503 A135468 A003196 this_sequence A072325 A076948 A086071
%K A062977 nonn
%O A062977 1,24
%A A062977 Henry Bottomley (se16(AT)btinternet.com), Jul 24 2001

%I A072325
%S A072325 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,1,0,
%T A072325 0,0,0,0,0,0,1,0,0,0,1,2,1,1,0,0,0,1,1,1,1,0,0,0,0,1,2,0,0,0,2,2,2,1,0,
%U A072325 0,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0,0,0,1,1,0,1,0,0,1,0,2,2,2,1,1,0,0
%N A072325 Number of even numbers that cannot be expressed as the difference p-q of two odd primes q < p <= prime(n).
%C A072325 If a(n)=0, then Prime[n], called a cluster prime, is in A038134. If a(n)>0 then Prime[n] is in A038133.
%H A072325 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ClusterPrime.html">Cluster Primes</a>
%e A072325 a(25)=1 because Prime[25]=97 and there is 1 even number, 88, that cannot be written as the difference of two odd primes less than or equal to 97.
%t A072325 m=10000; n=PrimePi[m]-1; p=Table[Prime[i+1], {i, n}]; d=Table[0, {m/2}]; c=Table[0, {n}]; For[i=2, i<=n, i++, For[j=1, j<i, j++, diff=p[[i]]-p[[j]]; d[[diff/2]]++ ]; c[[i]]=Count[Take[d, (p[[i]]-3)/2], 0]]; c
%Y A072325 Cf. A038133, A038134.
%Y A072325 Adjacent sequences: A072322 A072323 A072324 this_sequence A072326 A072327 A072328
%Y A072325 Sequence in context: A135468 A003196 A062977 this_sequence A076948 A086071 A089813
%K A072325 easy,nonn
%O A072325 2,30
%A A072325 T. D. Noe (noe(AT)sspectra.com), Jul 15 2002, Nov 19 2006

%I A076948
%S A076948 1,1,0,0,1,0,0,0,0,1,0,0,2,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,5,0,0,0,0,5,0,
%T A076948 0,1,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,5,0,0,2,0,0,0,1,0,0,0,0,
%U A076948 0,0,0,10,13,0,0,0,0,0,0,0,1,0,0,2,0,0,0,13,0,0,0,0,0,0,0,5,0,0,0,1,0
%N A076948 Smallest k such that nk-1 is a square, or 0 if no such number exists.
%t A076948 Do[k = 1; While[k < 10^4 && !IntegerQ[Sqrt[n*k - 1]], k++ ]; If[k != 10^4, Print[k], Print[0]], {n, 1, 105}]
%Y A076948 Cf. A008784.
%Y A076948 Adjacent sequences: A076945 A076946 A076947 this_sequence A076949 A076950 A076951
%Y A076948 Sequence in context: A003196 A062977 A072325 this_sequence A086071 A089813 A037845
%K A076948 nonn
%O A076948 1,13
%A A076948 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 20 2002
%E A076948 Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 21 2002

%I A086071
%S A086071 1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,1,0,0,0,2,0,0,0,1,0,0,0,0,0,0,
%T A086071 1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,1,2,0,0,0,0,0,
%U A086071 0,0,0,1,0,1,0,2,1,1,1,0,1,0,1,0,0,0,0,2,1,0,0,0,0,1,0,0,0,1,2,0,0,0,1
%N A086071 Number of 0's in decimal expansion of triangular number n(n+1)/2.
%e A086071 tri(4)=10, so a(4)=1 and tri(24)=300, so a(24)=2.
%Y A086071 Cf. 1's A086072, 2's A086073, 3's A086074, 4's A086075, 5's A086076, 6's A086077, 7's A086078, 8's A086079, 9's A086080.
%Y A086071 Adjacent sequences: A086068 A086069 A086070 this_sequence A086072 A086073 A086074
%Y A086071 Sequence in context: A062977 A072325 A076948 this_sequence A089813 A037845 A037881
%K A086071 base,nonn
%O A086071 0,25
%A A086071 Jason Earls (jcearls(AT)cableone.net), Jul 08 2003

%I A089813
%S A089813 1,0,2,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%T A089813 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A089813 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0
%V A089813 1,0,-2,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
%W A089813 1,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
%X A089813 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0
%N A089813 Expansion of Jacobi theta function (theta_2(q)-3theta_2(q^9))/2.
%D A089813 I. J. Zucker, "Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums." J. Phys. A: Math. Gen. 23, 117-132, 1990.
%H A089813 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%Y A089813 Adjacent sequences: A089810 A089811 A089812 this_sequence A089814 A089815 A089816
%Y A089813 Sequence in context: A072325 A076948 A086071 this_sequence A037845 A037881 A024326
%K A089813 sign
%O A089813 0,3
%A A089813 Eric Weisstein (eric(AT)weisstein.com), Nov 12, 2003

%I A037845
%S A037845 0,0,0,0,0,1,2,0,0,0,1,0,0,0,0,0,1,2,3,0,0,1,2,1,1,1,2,2,2,2,
%T A037845 2,0,1,2,3,0,0,1,2,0,0,0,1,1,1,1,1,0,1,2,3,0,0,1,2,0,0,0,1,0,
%U A037845 0,0,0,0,1,2,3,1,1,2,3,2,2,2,3,3,3,3,3,0,1,2,3,0,0,1,2,1,1,1
%N A037845 Sum{d(i)-d(i-1): d(i)>d(i-1), i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is base 4 representation of n.
%Y A037845 Adjacent sequences: A037842 A037843 A037844 this_sequence A037846 A037847 A037848
%Y A037845 Sequence in context: A076948 A086071 A089813 this_sequence A037881 A024326 A133698
%K A037845 nonn,base
%O A037845 1,7
%A A037845 Clark Kimberling (ck6(AT)evansville.edu)

%I A037881
%S A037881 0,0,0,0,0,1,2,0,0,0,1,0,0,0,0,0,1,2,3,0,0,1,2,1,1,1,2,2,2,2,
%T A037881 2,0,1,2,3,0,0,1,2,0,0,0,1,1,1,1,1,0,1,2,3,0,0,1,2,0,0,0,1,0,
%U A037881 0,0,0,0,1,2,3,1,1,2,3,2,2,3,4,3,3,4,5,0,1,2,3,0,0,1,2,1,1,2
%N A037881 (1/2)*Sum{|d(i)-e(i)|} where Sum{d(i)*4^i) is base 4 representation of n and e(i) are digits d(i) in nonincreasing order.
%Y A037881 Adjacent sequences: A037878 A037879 A037880 this_sequence A037882 A037883 A037884
%Y A037881 Sequence in context: A086071 A089813 A037845 this_sequence A024326 A133698 A093956
%K A037881 nonn,base
%O A037881 1,7
%A A037881 Clark Kimberling (ck6(AT)evansville.edu)

%I A024326
%S A024326 0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0,1,0,
%T A024326 0,2,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,1,0,0,1,0,0,0,0,2,0,0,0,0,0,
%U A024326 1,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1
%N A024326 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k=[ (n+1)/2) ], s = A023531, t = A023533.
%Y A024326 Adjacent sequences: A024323 A024324 A024325 this_sequence A024327 A024328 A024329
%Y A024326 Sequence in context: A089813 A037845 A037881 this_sequence A133698 A093956 A101436
%K A024326 nonn
%O A024326 1,39
%A A024326 Clark Kimberling (ck6(AT)evansville.edu)

%I A133698
%S A133698 1,0,1,0,0,2,0,0,0,1,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,
%T A133698 1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2
%N A133698 Triangle, diagonal = A001227 with the rest zeros.
%F A133698 Infinite lower triangular matrix with A001227 (number of odd divisors of n) as the main diagonal the rest zeros.
%e A133698 First few rows of the triangle are:
%e A133698 1;
%e A133698 0, 1;
%e A133698 0, 0, 2
%e A133698 0, 0, 0, 1;
%e A133698 0, 0, 0, 0, 2;
%e A133698 ...
%Y A133698 Cf. A133698.
%Y A133698 Adjacent sequences: A133695 A133696 A133697 this_sequence A133699 A133700 A133701
%Y A133698 Sequence in context: A037845 A037881 A024326 this_sequence A093956 A101436 A056170
%K A133698 nonn,tabl
%O A133698 0,6
%A A133698 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 21 2007

%I A093956
%S A093956 0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0,1,0,
%T A093956 0,0,1,1,1,1,2,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,
%U A093956 0,1,0,0,0,1,0,0,0,1,1,1,1,2,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0
%N A093956 A091787(n) - 2.
%C A093956 The purpose of A093955-A093958 is to compare A090822, A091787, A091799 and A091844 on the same "scale".
%C A093956 Sequence is unbounded.
%H A093956 F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and A. R. Wilks, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Slow-Growing Sequence Defined by an Unusual Recurrence</a>, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
%H A093956 F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and A. R. Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [<a href="http://www.research.att.com/~njas/doc/gijs.pdf">pdf</a>, <a href="http://www.research.att.com/~njas/doc/gijs.ps">ps</a>].
%Y A093956 Cf. A090822, A093955-A093958.
%Y A093956 Adjacent sequences: A093953 A093954 A093955 this_sequence A093957 A093958 A093959
%Y A093956 Sequence in context: A037881 A024326 A133698 this_sequence A101436 A056170 A059483
%K A093956 nonn
%O A093956 1,42
%A A093956 njas, May 22 2004

%I A101436
%S A101436 0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,1,0,0,0,
%T A101436 2,0,0,0,1,0,0,0,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,0,0,0,0,1,0,0,
%U A101436 0,2,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,1,1,2,0,0,0,1,0
%N A101436 Number of exponents in prime factorization of n which are primes.
%C A101436 First occurrence of k: 1,4,36,900,44100 (A061742). - Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 25 2005
%e A101436 36 = 2^2 *3^2. Since 2 is a prime and occurs twice as an exponent in the prime factorization of 36, a(36) = 2.
%t A101436 f[n_] := Length[ Select[ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]], PrimeQ[ # ] &]]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Jan 25 2005)
%Y A101436 Adjacent sequences: A101433 A101434 A101435 this_sequence A101437 A101438 A101439
%Y A101436 Sequence in context: A024326 A133698 A093956 this_sequence A056170 A059483 A067618
%K A101436 nonn
%O A101436 1,36
%A A101436 Leroy Quet (qq-quet(AT)mindspring.com), Jan 18 2005
%E A101436 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 25 2005

%I A056170
%S A056170 0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,1,0,0,0,
%T A056170 2,0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,
%U A056170 0,2,0,0,1,1,0,0,0,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,1,1,2,0,0,0,1,0
%N A056170 Number of non-unitary prime divisors of n.
%C A056170 Number of prime squares dividing n. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 18 2002
%F A056170 A prime factor of n is not unitary iff its exponent is p in prime factorization of n. In general GCD[p, n/p]=1 or =p
%F A056170 Additive with a(p^e) = 0 if e = 1, 1 otherwise.
%Y A056170 A034444, A001221.
%Y A056170 Cf. A057427(a(n)) = 1 - A008966(n).
%Y A056170 Adjacent sequences: A056167 A056168 A056169 this_sequence A056171 A056172 A056173
%Y A056170 Sequence in context: A133698 A093956 A101436 this_sequence A059483 A067618 A055029
%K A056170 nice,nonn
%O A056170 1,36
%A A056170 Labos E. (labos(AT)ana.sote.hu), Jul 27 2000

%I A059483
%S A059483 0,0,0,0,2,0,0,0,1,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,2,1,0,0,0,0,0,
%T A059483 0,8,0,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,2,18,
%U A059483 0,0,0,0,0,0,0,0,0,0,16,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A059483 Triangle T(n,k) = number of fixed 2 X n polyominoes with k cells = coefficient of x^n*y^k in (1+x*y)/(1-x*y-x*y^2-x^2*y^3)-1, read by rows in order 00, 10, 01, 20, 11, 02, ...
%D A059483 R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.
%H A059483 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%p A059483 read transforms; (1+x*y)/(1-x*y-x*y^2-x^2*y^3)-1; SERIES2(%,x,y,12): SERIES2TOLIST(%,x,y,12);
%Y A059483 Adjacent sequences: A059480 A059481 A059482 this_sequence A059484 A059485 A059486
%Y A059483 Sequence in context: A093956 A101436 A056170 this_sequence A067618 A055029 A126812
%K A059483 nonn,tabl,easy
%O A059483 0,5
%A A059483 njas, Feb 04 2001

%I A067618
%S A067618 1,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,1,0,0,1,2,0,0,0,1,0,0,0,2,0,
%T A067618 0,0,1,0,0,1,1,0,0,0,2,0,0,1,4,0,0,0,2,0,0,0,3,0,0,0,3,0,0,1,3,0,0,0,5,
%U A067618 0,0,1,6,0,0,0,3,0,0,0,5,0,0,0,6,0,0,1,5,0,0,0,7,0,0,0,9,0,0,0,5,0
%N A067618 Number of self-conjugate partitions of n into prime parts.
%t A067618 f[0, m_, k_] := 1; f[n_, 0, k_] := If[n==0, 1, 0]; f[n_, m_, k_] := If[n<0||m<0, 0, Module[{r}, f[n, m, k]=f[n, m-1, k]+If[PrimeQ[m+k], Sum[If[PrimeQ[r+k], f[n-r(2m-r), m-r-1, k+r], 0], {r, 1, m}], 0]]]; a[n_] := f[n, Floor[n/4]+1, 0]; (* f[n, m, k] = number of self-conjugate partitions of n with parts <= m such that every part+k is prime *)
%Y A067618 Cf. A000700, A000701, A046682.
%Y A067618 Adjacent sequences: A067615 A067616 A067617 this_sequence A067619 A067620 A067621
%Y A067618 Sequence in context: A101436 A056170 A059483 this_sequence A055029 A126812 A008442
%K A067618 easy,nonn
%O A067618 0,26
%A A067618 Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 01 2002
%E A067618 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Feb 11 2002

%I A055029
%S A055029 0,0,1,0,0,2,0,0,0,1,0,0,0,2,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,
%T A055029 0,0,2,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,
%U A055029 0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0
%N A055029 Number of inequivalent Gaussian primes of norm n.
%C A055029 These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).
%C A055029 Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-i).
%D A055029 R. K. Guy, Unsolved Problems in Number Theory, A16.
%D A055029 L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
%H A055029 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ga.html#gaussians">Index entries for Gaussian integers and primes</a>
%F A055029 a(n) = 2 if n is a prime = 1 (mod 4); a(n) = 1 if n is 2, or p^2 where p is a prime = 3 (mod 4); a(n) = 0 otherwise. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 05 2006
%e A055029 There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i).
%Y A055029 Cf. A055025-A055028, A055664-...
%Y A055029 Adjacent sequences: A055026 A055027 A055028 this_sequence A055030 A055031 A055032
%Y A055029 Sequence in context: A056170 A059483 A067618 this_sequence A126812 A008442 A086076
%K A055029 nonn,easy,nice
%O A055029 0,6
%A A055029 njas, Jun 09 2000
%E A055029 More terms from Reiner Martin (reinermartin(AT)hotmail.com), Jul 20 2001

%I A126812
%S A126812 1,0,0,0,2,0,0,0,1,0,0,0,2,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,2,0,0,0,0,0,0,
%T A126812 0,2,0,0,0,2,0,0,0,2,0,0,0,1,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,
%U A126812 0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0
%N A126812 Ramanujan numbers (A000594) read mod 4.
%D A126812 H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients, of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
%Y A126812 Cf. A000594.
%Y A126812 Adjacent sequences: A126809 A126810 A126811 this_sequence A126813 A126814 A126815
%Y A126812 Sequence in context: A059483 A067618 A055029 this_sequence A008442 A086076 A085981
%K A126812 nonn
%O A126812 1,5
%A A126812 njas, Feb 25 2007

%I A008442
%S A008442 1,0,0,0,2,0,0,0,1,0,0,0,2,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,2,0,0,0,0,0,0,
%T A008442 0,2,0,0,0,2,0,0,0,2,0,0,0,1,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,4,0,0,0,0,0,
%U A008442 0,0,2,0,0,0,0,0,0,0,1,0,0,0,4,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0
%N A008442 Expansion of Jacobi theta constant (theta_2(2z))^2/4.
%C A008442 Expansion of eta(q^8)^4/eta(q^4)^2 in powers of q.
%C A008442 Euler transform of period 8 sequence [0,0,0,2,0,0,0,-2,...]. - Michael Somos Apr 24 2004
%C A008442 a(n)=0 unless n=4k+1 in which case a(n) is the difference between number of divisors of n of form 4k+1 and 4k+3.
%C A008442 Multiplicative with a(2^e)=0 if e>0, a(p^e)=(1+(-1)^e)/2 if p==3 mod 4 otherwise a(p^e)=1+e. - Michael Somos Sep 18 2004
%D A008442 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%D A008442 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.26).
%H A008442 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b008442.txt">Table of n, a(n) for n = 1..1000</a>
%F A008442 Fine gives an explicit formula for a(n) in terms of the divisors of n.
%F A008442 a(n) = number of divisors of n of form 8n+1, 8n+5, 8n+6 minus number of divisors of form 8n+2, 8n+3, 8n+7. [I think Fine's version is simpler - njas]
%F A008442 G.f. = s(8)^4/(s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
%F A008442 Moebius transform is period 8 sequence [1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos Sep 02 2005
%F A008442 G.f.: Sum_{k>0} kronecker(-4, k) x^k/(1-x^(2k)) = Sum_{k>0} x^(2k-1)/(1+x^(4k-2)) . - Michael Somos Sep 20 2005
%F A008442 G.f.: Sum_{k>0} x^k(1-x^k)(1-x^(2k))(1-x^(3k))/(1-x^(8k)) = x Product_{k>0} (1-x^(8k))^4/(1-x^(4k))^2. - Michael Somos Apr 24 2004
%F A008442 Moebius transform is period 8 sequence [1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Sep 02 2005
%o A008442 (PARI) a(n)=if(n<1|n%4!=1,0,sumdiv(n,d,(d%4==1)-(d%4==3))) /* Michael Somos Apr 24 2004 */
%o A008442 (PARI) a(n)=if(n<1,0,sumdiv(n,d,[0,1,-1,-1,0,1,1,-1][d%8+1])) /* Michael Somos Apr 24 2004 */
%o A008442 (PARI) a(n)=local(A); if(n<1,0,n--; A=x*O(x^n); polcoeff(eta(x^8+A)^4/eta(x^4+A)^2,n)) /* Michael Somos Apr 24 2004 */
%Y A008442 A008441(n)=a(4n+1).
%Y A008442 Adjacent sequences: A008439 A008440 A008441 this_sequence A008443 A008444 A008445
%Y A008442 Sequence in context: A067618 A055029 A126812 this_sequence A086076 A085981 A127324
%K A008442 nonn,mult
%O A008442 1,5
%A A008442 njas

%I A086076
%S A086076 0,0,0,0,0,1,0,0,0,1,2,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,1,0,0,1,1,0,1,1,2,
%T A086076 0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0,1,
%U A086076 1,2,0,0,1,1,0,0,0,0,0,0,0,0,1,2,0,0,0,1,1,0,0,0,1,1,1,1,1,1,2,2,2,2,1
%N A086076 Number of 5's in decimal expansion of triangular number n(n+1)/2.
%Y A086076 Cf. 0's A086071, 1's A086072, 2's A086073, 3's A086074, 4's A086075, 6's A086077, 7's A086078, 8's A086079, 9's A086080.
%Y A086076 Adjacent sequences: A086073 A086074 A086075 this_sequence A086077 A086078 A086079
%Y A086076 Sequence in context: A055029 A126812 A008442 this_sequence A085981 A127324 A083917
%K A086076 base,nonn
%O A086076 0,11
%A A086076 Jason Earls (jcearls(AT)cableone.net), Jul 08 2003

%I A085981
%S A085981 0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,0,
%T A085981 0,1,0,1,1,1,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,1,2,0,0,0,1,0,0,1,0,1,1,0,
%U A085981 0,0,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1
%N A085981 Number of 7's in decimal expansion of prime(n).
%Y A085981 Cf. 0's A085974, 1's A085975, 2's A085976, 3's A085977, 4's A085978, 5's A085979, 6's A085980, 8's A085982, 9's A085983.
%Y A085981 Adjacent sequences: A085978 A085979 A085980 this_sequence A085982 A085983 A085984
%Y A085981 Sequence in context: A126812 A008442 A086076 this_sequence A127324 A083917 A117974
%K A085981 base,nonn
%O A085981 1,59
%A A085981 Jason Earls (jcearls(AT)cableone.net), Jul 06 2003

%I A127324
%S A127324 0,0,0,0,1,0,0,0,1,0,0,1,0,1,2,0,0,0,1,0,0,1,0,1,2,0,0,1,0,1,2,0,1,2,3,
%T A127324 0,0,0,1,0,0,1,0,1,2,0,0,1,0,1,2,0,1,2,3,0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,
%U A127324 0,0,0,1,0,0,1,0,1,2,0,0,1,0,1,2,0,1,2,3,0,0,1,0,1,2,0,1,2,3,0,1,2,3,4
%N A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.
%C A127324 If {(W,X,Y,Z)} are 4-tuples of nonnegative integers with W>=X>=Y>=Z ordered by W, X, Y, and Z, then W=A127321(n), X=A127322(n), Y=A127323(n) and Z=A127324(n). These sequences are the four-dimensional analogues of the three-dimensional A056556, A056557, and A056558.
%F A127324 For W>=X>=Y>=Z>=0, a(A000332(W+3)+A000292(X)+A000217(Y)+Z) = Z A127322(n+1) = A127321(n)==A127324(n) ? 0 : A127322(n)==A127324(n) ? 0 : A127323(n)==A127324(n) ? 0 : A127324(n)+1
%e A127324 See A127321 for a table of A127321, A127322, A127323, A127324
%e A127324 See A127327 for a table of A127324, A127325, A127326, A127327
%Y A127324 Cf. A127321, A127322, A127323, A056556, A056557, A056558, A000332, A000292, A000217.
%Y A127324 Adjacent sequences: A127321 A127322 A127323 this_sequence A127325 A127326 A127327
%Y A127324 Sequence in context: A008442 A086076 A085981 this_sequence A083917 A117974 A085983
%K A127324 nonn
%O A127324 0,15
%A A127324 Graeme McRae (g_m(AT)mcraefamily.com), Jan 10 2007

%I A083917
%S A083917 0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,1,
%T A083917 0,1,0,0,0,0,1,0,0,0,0,1,0,1,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,0,0,1,1,0,1,
%U A083917 0,0,0,1,0,0,2,0,0,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,1,0,0,0,1,0,0,1
%N A083917 Number of divisors of n that are congruent to 7 modulo 10.
%C A083917 a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083918(n) - A083919(n).
%Y A083917 Cf. A010879, A000005, A001227.
%Y A083917 Adjacent sequences: A083914 A083915 A083916 this_sequence A083918 A083919 A083920
%Y A083917 Sequence in context: A086076 A085981 A127324 this_sequence A117974 A085983 A088183
%K A083917 nonn
%O A083917 1,77
%A A083917 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 08 2003

%I A117974
%S A117974 1,0,0,1,1,2,0,0,0,1,0,0,1,1,3,0,0,1,0,2,1,0,0,0,1,2,1,4,0,0,0,1,1,1,3,
%T A117974 3,0,0,1,0,2,1,4,2,5,0,0,0,1,1,0,2,2,2,3,0,0,0,0,1,1,3,2,4,2,7,0,0,0,0,
%U A117974 0,1,2,1,2,1,5,2,0,0,0,1,2,0,3,2,3,3,7,2,9,0,0,0,0,1,1,3,1,3,0,5,2,7,3
%N A117974 Triangle where a(1,1)=1; a(n,m) = number of positive integers which are missing from row (n-1) of the triangle, are <= m, and are coprime to m.
%e A117974 Row 5 of the triangle is [0,0,1,1,3]. There are 0 positive integers which are coprime to 1, are <= 1, and are not among the terms of row 5 (because 1 occurs in row 5). There are 0 positive integers which are <= 2, are coprime to 2, and are not among the terms of row 5. ...(Skipping over the m = 3, 4, and 5 cases.) There is 1 positive integer (5) which is <= 6, is coprime to 6, and does not occur in row 5.
%e A117974 So row 6 is [0,0,1,0,2,1].
%p A117974 A117974 := proc(nrow) local a,aprev,anm,m,k ; if nrow = 1 then [1] ; else a := [] ; aprev := A117974(nrow-1) ; for m from 1 to nrow do anm := 0 ; for k from 1 to m do if not k in aprev and gcd(k,m) = 1 then anm := anm+1 ; fi ; od: a := [op(a),anm] ; od; RETURN(a) ; fi ; end: seq(op(A117974(n)),n=1..20) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 05 2007
%Y A117974 Cf. A117975.
%Y A117974 Adjacent sequences: A117971 A117972 A117973 this_sequence A117975 A117976 A117977
%Y A117974 Sequence in context: A085981 A127324 A083917 this_sequence A085983 A088183 A070140
%K A117974 nonn,tabl
%O A117974 1,6
%A A117974 Leroy Quet (qq-quet(AT)mindspring.com), Apr 06 2006
%E A117974 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 05 2007

%I A085983
%S A085983 0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,1,1,
%T A085983 0,0,0,0,0,1,0,1,1,1,2,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,
%U A085983 0,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0,0,0,1,0,1,2,0,1,0,0,0,0,0,0,1,0
%N A085983 Number of 9's in decimal expansion of prime(n).
%Y A085983 Cf. 0's A085974, 1's A085975, 2's A085976, 3's A085977, 4's A085978, 5's A085979, 6's A085980, 7's A085981, 8's A085982.
%Y A085983 Adjacent sequences: A085980 A085981 A085982 this_sequence A085984 A085985 A085986
%Y A085983 Sequence in context: A127324 A083917 A117974 this_sequence A088183 A070140 A081212
%K A085983 base,nonn
%O A085983 1,46
%A A085983 Jason Earls (jcearls(AT)cableone.net), Jul 06 2003

%I A088183
%S A088183 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,1,0,1,0,0,0,2,0,3,0,0,1,1,
%T A088183 0,2,0,2,0,2,0,4,1,0,1,4,0,2,0,1,0,3,0,4,0,1,2,5,0,6,0,1,3,1,0,4,1,3,0,
%U A088183 6,0,5,3,1,2,3,0,5,0,3,2,7,0,1,3,4,1,4,0,6,2,2,3,6,0,7,1,4,2,6,1
%N A088183 Number of ways to write n as a sum of two coprime semiprimes.
%C A088183 a(A088184(n))>0, a(A088185(n))=0;
%C A088183 is a(n)>0 for n>210? see conjecture in A072931.
%C A088183 The graph of this sequence is compelling evidence that 210 is the last term of sequence A088185. - T. D. Noe, Apr 10 2007
%H A088183 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b088183.txt">Table of n, a(n) for n=1..10000</a>
%H A088183 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>
%H A088183 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RelativelyPrime.html">Relatively Prime</a>
%e A088183 a(64)=3: 64 = 3*3+5*11 = 3*5+7*7 = 5*5+3*13, (A072931(64)=5).
%Y A088183 Cf. A072931, A001358.
%Y A088183 Adjacent sequences: A088180 A088181 A088182 this_sequence A088184 A088185 A088186
%Y A088183 Sequence in context: A083917 A117974 A085983 this_sequence A070140 A081212 A085491
%K A088183 nonn
%O A088183 1,19
%A A088183 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 22 2003

%I A070140
%S A070140 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%T A070140 1,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,1,0,1,
%U A070140 0,1,0,0,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1
%N A070140 Number of acute integer triangles with perimeter n having integral area.
%C A070140 a(n) = A051516(n) - A070141(n) - A024155(n).
%H A070140 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%H A070140 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AcuteTriangle.html">Acute Triangle</a>.
%H A070140 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a070080.txt">Integer-sided triangles</a>
%Y A070140 Cf. A051516, A070093, A070146.
%Y A070140 Adjacent sequences: A070137 A070138 A070139 this_sequence A070141 A070142 A070143
%Y A070140 Sequence in context: A117974 A085983 A088183 this_sequence A081212 A085491 A116681
%K A070140 nonn
%O A070140 1,64
%A A070140 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002

%I A081212
%S A081212 0,0,0,1,0,0,0,1,1,0,0,2,0,0,0,1,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,1,0,0,0,
%T A081212 1,0,0,0,1,0,0,0,1,1,0,0,2,1,1,0,1,0,2,0,2,0,0,0,2,0,0,2,1,0,0,0,1,0,0,
%U A081212 0,2,0,0,1,1,0,0,0,2,1,0,0,3,0,0,0,1,0,1,0,1,0,0,0,1,0,1,1,1,0,0,0,1,0
%N A081212 Let r(n,k) = if k=0 then n else r(A081210(n),k-1), then a(n)=Min{i:r(n,i)=r(n,i+1)}.
%C A081212 Number of times A081210 is to be applied to n until a x-point is reached, A081213(n)=r(n,a(n)).
%C A081212 A081211(n) = A081213(n) iff a(n) <= 2; a(A131072(n)) > 2. - Reinhard Zumkeller (reinhard.zumkeller@gmail.com), Jun 13 2007
%H A081212 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/b081212.txt">Table of n, a(n) for n = 0..10000</a>
%Y A081212 Adjacent sequences: A081209 A081210 A081211 this_sequence A081213 A081214 A081215
%Y A081212 Sequence in context: A085983 A088183 A070140 this_sequence A085491 A116681 A131371
%K A081212 nonn
%O A081212 1,12
%A A081212 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 10 2003

%I A085491
%S A085491 1,0,1,0,1,0,1,0,0,0,2,0,0,0,1,0,1,0,1,0,0,0,5,0,0,0,1,0,3,0,1,0,0,0,5,
%T A085491 0,0,0,3,0,2,0,0,0,0,0,10,0,0,0,0,0,1,0,2,0,0,0,31,0,0,0,1,0,1,0,0,0,1,
%U A085491 0,26,0,0,0,0,0,1,0,6,0,0,0,23,0,0,0,1,0,20,0,0,0,0,0,21,0,0,0,1
%N A085491 Number of ways to write n as sum of distinct divisors of n+1.
%C A085491 a(A085492(n)) = 0; a(A085493(n)) > 0; a(A085494(n)) = 1.
%e A085491 n=11, divisors of 12=11+1 that are not greater 11: {1,2,3,4,6},
%e A085491 11=6+5=6+4+1, therefore a(11)=2.
%Y A085491 Cf. A085496.
%Y A085491 Adjacent sequences: A085488 A085489 A085490 this_sequence A085492 A085493 A085494
%Y A085491 Sequence in context: A088183 A070140 A081212 this_sequence A116681 A131371 A003475
%K A085491 nonn
%O A085491 1,11
%A A085491 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 03 2003

%I A116681
%S A116681 1,0,1,1,0,0,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,2,0,0,0,1,0,1,0,2,0,1,0,1,0,
%T A116681 1,2,0,0,0,1,0,1,0,2,0,2,0,2,0,1,0,1,0,2,3,0,0,0,2,0,1,0,2,0,2,0,3,0,2,
%U A116681 0,2,0,1,0,2,0,2,4,0,0,0,2,0,2,0,2,0,2,0,3,0,4,0,3,0,2,0,2,0,2,0,2,0,3
%N A116681 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, in which the sum of the odd parts is k (n>=0, 0<=k<=n).
%C A116681 Row sums yield A000009. T(2n,0)=A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n+1,1)=A000009(n), T(n,2)=0. T(n,n)=A000700(n). Sum(k*T(n,k), k=0..n)=A116682(n).
%F A116681 G.f.=product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).
%e A116681 T(10,4)=2 because we have [6,3,1] and [4,3,2,1].
%e A116681 Triangle starts:
%e A116681 1;
%e A116681 0,1;
%e A116681 1,0,0;
%e A116681 0,1,0,1;
%e A116681 1,0,0,0,1;
%e A116681 0,1,0,1,0,1;
%p A116681 g:=product((1+(t*x)^(2*j-1))*(1+x^(2*j)),j=1..30): gser:=simplify(series(g,x=0,20)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 15 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y A116681 Cf. A000009, A035457, A036469, A116676.
%Y A116681 Adjacent sequences: A116678 A116679 A116680 this_sequence A116682 A116683 A116684
%Y A116681 Sequence in context: A070140 A081212 A085491 this_sequence A131371 A003475 A135767
%K A116681 nonn,tabl
%O A116681 0,22
%A A116681 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2006

%I A131371
%S A131371 0,0,0,1,0,1,0,0,1,1,0,1,0,1,2,0,0,0,1,0,1,1,0,0,1,2,0,1,0,0,0,0,1,1,1,
%T A131371 0,0,1,2,0,1,0,1,0,0,1,1,0,2,0,2,1,1,0,1,1,1,2,1,0,0,2,0,1,1,0,0,1,1,0,
%U A131371 0,0,0,1,1,0,1,1,0,0,0,1,1,0,2,1,1,0,0,0,1,0,2,2,1,1,0,0,0,0,0,1,1,0,1
%N A131371 Number of anagrams of n that are semiprimes.
%C A131371 An anagram of a k-digit number is one of the k! permutations of the digits that does not begin with 0. This is to semiprimes A001358 as A046810 is to primes A000040.
%e A131371 a(123) = 3 because 123 = 3 * 41 is semiprime, 213 = 3 * 71 is semiprime, 321 = 3 * 107 is semiprime, while the other anagrams 132, 231, and 312 have respectively 3, 3, and 5 prime factors with multiplicity.
%e A131371 a(129) = 4 because 129 = 3 * 43 is semiprime, 219 = 3 * 73 is semiprime, 291 = 3 * 97 is semiprime, 921 = 3 * 307 is semiprime, while 192 and 912 have 7 and 6 prime factors with multiplicity.
%e A131371 a(134) = 5 because 134 = 2 * 67, and 143 = 11 * 13, and 314 = 2 * 157, and 341 = 11 * 31, and 413 = 7 * 59 are semiprimes, while 431 is prime.
%Y A131371 Cf. A000040, A001358, A002113, A046810, A097393.
%Y A131371 Adjacent sequences: A131368 A131369 A131370 this_sequence A131372 A131373 A131374
%Y A131371 Sequence in context: A081212 A085491 A116681 this_sequence A003475 A135767 A070203
%K A131371 base,easy,nonn
%O A131371 1,15
%A A131371 Jonathan Vos Post (jvospost2(AT)yahoo.com), Sep 30 2007

%I A003475
%S A003475 1,1,1,0,0,0,1,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0,1,
%T A003475 1,1,1,0,0,1,0,1,0,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0,1,1,0,1,1,1,0,0,2,
%U A003475 0,0,0,1,0,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,1,2,0,0,1,0,1,0,1,1,1,0,0,0
%V A003475 1,1,1,0,0,0,-1,-1,0,-1,-1,0,-1,0,1,-1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,-1,0,0,0,1,-1,-1,-1,
%W A003475 0,0,-1,0,-1,0,0,-1,-1,-1,1,0,-1,0,0,0,1,0,-1,0,0,1,0,1,1,0,1,-1,1,0,0,2,0,0,0,1,0,1,1,
%X A003475 -1,0,0,0,0,0,0,1,-1,-1,1,0,0,0,-1,-2,0,0,-1,0,1,0,-1,-1,-1,0,0,0
%N A003475 Expansion of Sum { (-1)^n q^(n^2) / ((1-q)(1-q^3)(1-q^5)...(1-q^(2n-1))) }; n=0..inf.
%C A003475 |a(n)|<3 if n<1036, a(1036)=3. - Michael Somos Sep 16 2006
%D A003475 F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
%D A003475 F. J. Dyson, Selected Papers, Am. Math. Soc., 1996, p. 204.
%F A003475 Define c(24k+1)=A003406(k), c(24k-1)=-2*A003475(k), c(n)=0 otherwise. Then c(n) is multiplicative with c(2^e)=c(3^e)=0^e, c(p^e) = (-1)^(e/2)*(1+(-1)^e)/2 if p == 7, 17 (mod 24), c(p^e) = (1+(-1)^e)/2 if p == 5, 11, 13, 19 (mod 24), c(p^e) = (e+1)*(-1)^(y*e) where p == 1, 23 (mod 24) and p = x^2-72*y^2 . - Michael Somos Aug 17 2006 */
%F A003475 G.f.: x +x^2(1-x^2) +x^3(1-x^2)(1-x^4) +x^4(1-x^2)(1-x^2)(1-x^6) +... . - Michael Somos Aug 18 2006
%o A003475 (PARI) {a(n)=local(A, p, e, x, y); if(n<0, 0, n=24*n-1; A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%24>1&p%24<23, if(e%2, 0, if(p%24==7|p%24==17, (-1)^(e/2), 1)), x=y=0; if(p%24==1, forstep(i=1, sqrtint(p), 2, if(issquare((i^2+p)/2,&y), x=i; break)), for(i=1,sqrtint(p\2), if(issquare(2*i^2+p,&x), y=i; break))); (e+1)*(-1)^( (x+if((x-y)%6,y,-y))/6*e)))))/-2)} /* Michael Somos Aug 17 2006 */
%o A003475 (PARI) {a(n)=local(A); if(n<1, 0, A=-1+x*O(x^n); polcoeff( sum(k=1, sqrtint(n), A*= 1/(1-x^(1-2*k))*(1+x*O(x^(n-k^2)))), n))} /* Michael Somos Sep 16 2006 */
%Y A003475 Cf. A053251.
%Y A003475 Adjacent sequences: A003472 A003473 A003474 this_sequence A003476 A003477 A003478
%Y A003475 Sequence in context: A085491 A116681 A131371 this_sequence A135767 A070203 A070201
%K A003475 sign
%O A003475 1,70
%A A003475 njas

%I A135767
%S A135767 1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,1,0,2,0,0,0,0,0,
%T A135767 3,0,0,0,2,0,2,0,1,1,0,0,3,0,1,0,1,0,2,0,2,0,0,0,5,0,0,1,0,0,2,0,1,0,2,
%U A135767 0,5,0,0,1,1,0,2,0,3,0,0,0,5,0,0,0,2,0,5,0,1,0,0,0,4,0,1,1,3,0,2,0,2,2
%N A135767 sigma_0(n)-omega(n)-Omega(n) (sigma_0 = A000005 = # divisors, omega = A001221 = # prime factors, Omega = A001222 = # prime factors with multiplicity).
%C A135767 A102467 = { n | a(n)>0 } ; A102466 = { n | a(n)=0 } = { n | omega(n)=1 or Omega(n)=2 }: these are exactly the prime powers (>1) and semiprimes. For all other numbers a(n) > 0 since for each of the Omega(n) prime power divisors, other divisors are obtained by multiplying it with another prime factor, which gives more than omega(n) different additional divisors. a(n)>0 is also equivalent to A001037(n) > A107847(n), i.e. there are strictly fewer nonzero sums of non-periodic subsets of U_n (n-th roots of unity) than there are non-periodic binary words of length n. Otherwise stated, a(n)>0 if there is a non-periodic subset of U_n with zero sum. Non-periodic means having no rotational symmetry (except for identity).
%H A135767 M. F. Hasler, <a href="http://www.research.att.com/~njas/sequences/b135767.txt">Table of n, a(n) for n = 1..10000</a>
%F A135767 a(n)=0 <=> omega(n)=1 or Omega(n)=2 <=> n is semiprime or a prime power (>1) <=> A001037(n) = A107847(n) <=> all non-periodic subsets of U_n have nonzero sum
%o A135767 (PARI) A135767(n)=numdiv(n)-omega(n)-bigomega(n)
%Y A135767 Cf. A102466, A102467 ; A001037, A107847.
%Y A135767 Adjacent sequences: A135764 A135765 A135766 this_sequence A135768 A135769 A135770
%Y A135767 Sequence in context: A116681 A131371 A003475 this_sequence A070203 A070201 A070138
%K A135767 easy,nonn
%O A135767 1,24
%A A135767 M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jan 14 2008

%I A070203
%S A070203 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
%T A070203 2,0,0,0,1,0,2,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,3,0,0,0,1,0,1,0,1,0,2,
%U A070203 0,2,0,0,0,1,0,1,0,2,0,0,0,8,0,0,0,1,0,3
%N A070203 Number of scalene integer triangles with perimeter n having integral inradius.
%C A070203 a(n) = A070201(n) - A070204(n).
%H A070203 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Incircle.html">Incircle</a>.
%H A070203 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ScaleneTriangle.html">Scaline Triangle</a>.
%H A070203 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a070080.txt">Integer-sided triangles</a>
%Y A070203 Cf. A024153, A005044.
%Y A070203 Adjacent sequences: A070200 A070201 A070202 this_sequence A070204 A070205 A070206
%Y A070203 Sequence in context: A131371 A003475 A135767 this_sequence A070201 A070138 A024153
%K A070203 nonn
%O A070203 1,36
%A A070203 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002

%I A070201
%S A070201 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,
%T A070201 2,0,0,0,1,0,2,0,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,3,0,0,0,2,0,1,0,1,0,2,
%U A070201 0,2,0,0,0,1,0,1,0,2,0,0,0,8,0,0,0,1,0,3
%N A070201 Number of integer triangles with perimeter n having integral inradius.
%C A070201 a(n) = #{k | A070083(k) = n and A070200(k) = exact inradius};
%C A070201 a(n) = A070203(n) + A070204(n);
%C A070201 a(n) = A070205(n) + A070206(n) + A024155(n);
%C A070201 a(odd) = 0.
%H A070201 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Incircle.html">Incircle</a>.
%H A070201 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>.
%H A070201 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a070080.txt">Integer-sided triangles</a>
%e A070201 a(36)=2, as there are two integer triangles with integer inradius having perimeter=32:
%e A070201 First: [A070080(368), A070081(368), A070082(368)] = [9,10,17], for s=A070083(368)/2=(9+10+17)/2=18: inradius = SquareRoot((s-9)*(s-10)*(s-17)/s) = SquareRoot(9*8*1/18) = SquareRoot(4) = 2; therefore A070200(368)=2.
%e A070201 2nd: [A070080(370), A070081(370), A070082(370)] = [9,12,15], for s=A070083(370)/2=(9+12+15)/2=18: inradius = SquareRoot((s-9)*(s-12)*(s-15)/s) = SquareRoot(9*6*3/18) = SquareRoot(9) = 3; therefore A070200(370)=3.
%Y A070201 Cf. A070209, A070202, A070208, A005044, A070140.
%Y A070201 Adjacent sequences: A070198 A070199 A070200 this_sequence A070202 A070203 A070204
%Y A070201 Sequence in context: A003475 A135767 A070203 this_sequence A070138 A024153 A079127
%K A070201 nonn
%O A070201 1,36
%A A070201 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002

%I A070138
%S A070138 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,
%T A070138 0,1,0,0,0,2,0,0,0,1,0,2,0,1,0,0,0,1,0,2,0,0,0,3,0,1,0,0,0,1,
%U A070138 0,0,0,3,0,1,0,1,0,2,0,1,0,0,0,1,0,1,0,2,0,0,0,5,0,0,0,0,0,3
%N A070138 Number of integer triangles with an integer area having relatively prime sides a, b and c such that a+b+c=n.
%H A070138 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%H A070138 R. Zumkeller, <a href="http://www.research.att.com/~njas/sequences/a070080.txt">Integer-sided triangles</a>
%Y A070138 Cf. A051516, A070088, A070109, A070143.
%Y A070138 Adjacent sequences: A070135 A070136 A070137 this_sequence A070139 A070140 A070141
%Y A070138 Sequence in context: A135767 A070203 A070201 this_sequence A024153 A079127 A056674
%K A070138 nonn
%O A070138 1,36
%A A070138 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002
%E A070138 Corrected by T. D. Noe (noe(AT)sspectra.com), Jun 17 2004

%I A024153
%S A024153 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,
%T A024153 0,0,1,0,2,0,1,0,0,0,2,0,0,0,0,0,3,0,1,0,0,0,3,0,0,0,2,0,1,0,1,0,2,0,3,0,0,
%U A024153 0,1,0,1,0,3,0,0,0,8,0,0,0,1,0,4,0,0,0,0,0,4,0,3,0,0
%N A024153 Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n that have integer area.
%Y A024153 Adjacent sequences: A024150 A024151 A024152 this_sequence A024154 A024155 A024156
%Y A024153 Sequence in context: A070203 A070201 A070138 this_sequence A079127 A056674 A037188
%K A024153 nonn
%O A024153 1,36
%A A024153 Clark Kimberling (ck6(AT)evansville.edu)

%I A079127
%S A079127 0,0,0,0,0,0,2,0,0,0,1,0,2,0,1,2,2,1,5,3,2,2,6,6,6,2,4,4,1,3,3,4,2,5,7,
%T A079127 6,6,10,5,9,5,13,7,9,7,7,12,9,5,8,6,9,13,7,12,16,4,13,10,11,12,19,13,17,
%U A079127 12,10,9,14,14,15,15,14,19,12,26,14,18,11,11,18,18,22,8,23,22,11,21,14
%N A079127 Number of 5's in n# (n primorial) = 5's in A002110(n).
%o A079127 (PARI) See program in Number of 0's in n#
%Y A079127 Cf. A002110.
%Y A079127 Adjacent sequences: A079124 A079125 A079126 this_sequence A079128 A079129 A079130
%Y A079127 Sequence in context: A070201 A070138 A024153 this_sequence A056674 A037188 A086079
%K A079127 easy,nonn
%O A079127 2,7
%A A079127 Cino Hilliard (hillcino368(AT)gmail.com), Feb 03 2003

%I A056674
%S A056674 0,0,0,1,0,0,0,1,1,0,0,2,0,0,0,1,0,2,0,2,0,0,0,2,1,0,1,2,0,0,0,1,0,0,0,
%T A056674 3,0,0,0,2,0,0,0,2,2,0,0,2,1,2,0,2,0,2,0,2,0,0,0,4,0,0,2,1,0,0,0,2,0,0,
%U A056674 0,3,0,0,2,2,0,0,0,2,1,0,0,4,0,0,0,2,0,4,0,2,0,0,0,2,0,2,2,3,0,0,0,2,0
%N A056674 Number of square-free divisors which are not unitary. Also number of unitary divisors which are not square-free.
%C A056674 Numbers of unitary and of square-free divisors are identical, although the 2 sets are usually different, so sizes of parts outside overlap are also equal to each other.
%F A056674 a(n)=A034444(n)-A000005[A055231(n)] a(n)=A034444(n)-A000005[A007913(n)/A055229(n)]
%e A056674 n=252, it has 18 divisors, 8 are unitary, 8 are square-free, 2 belong to both classes, so 6 are square-free but not unitary, thus a(252)=6. Set {2,3,14,21,42} forms square-free but non-unitary while set {4,9,36,28,63,252} of same size gives the set of not square-free but unitary divisors.
%Y A056674 A034444, A000005, A055231, A007913, A055229 a(n)=A000005[A055231(n)]=A000005[A007913(n)/A055229(n)]
%Y A056674 Adjacent sequences: A056671 A056672 A056673 this_sequence A056675 A056676 A056677
%Y A056674 Sequence in context: A070138 A024153 A079127 this_sequence A037188 A086079 A133703
%K A056674 nonn
%O A056674 1,12
%A A056674 Labos E. (labos(AT)ana.sote.hu), Aug 10 2000

%I A037188
%S A037188 0,0,1,1,0,2,0,0,0,1,0,2,1,3,1,1,0,2,0,2,0,1,1,2,2,0,0,0,1,1,2,1,0,0,1,1,0,
%T A037188 1,0,0,0,1,0,1,0
%N A037188 Number of ways of placing 2n points on n X n grid so no 3 are in a line (solutions with symmetry about both main diagonals).
%D A037188 A. Flammenkamp, Progress in the no-three-in-line problem, J. Combinat. Theory A 60 (1992), 305-311.
%D A037188 A. Flammenkamp, Progress in the no-three-in-line problem. II. J. Combin. Theory Ser. A 81 (1998), no. 1, 108-113.
%H A037188 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/no3in/table_old.txt">Solutions of the no-three-in-line problem</a>
%H A037188 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/no3in/table.txt">Solutions of the no-three-in-line problem</a>
%Y A037188 Cf. A000769.
%Y A037188 Adjacent sequences: A037185 A037186 A037187 this_sequence A037189 A037190 A037191
%Y A037188 Sequence in context: A024153 A079127 A056674 this_sequence A086079 A133703 A073265
%K A037188 nonn
%O A037188 1,6
%A A037188 njas
%E A037188 More terms from Flammenkamp web site, May 24 2005

%I A086079
%S A086079 0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,
%T A086079 0,0,0,0,1,1,1,0,0,0,0,1,1,0,0,0,0,1,0,1,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,
%U A086079 1,0,1,0,0,1,0,0,1,0,0,0,0,1,0,0,0,2,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0
%N A086079 Number of 8's in decimal expansion of triangular number n(n+1)/2.
%Y A086079 Cf. 0's A086071, 1's A086072, 2's A086073, 3's A086074, 4's A086075, 5's A086076, 6's A086077, 7's A086078, 9's A086080.
%Y A086079 Adjacent sequences: A086076 A086077 A086078 this_sequence A086080 A086081 A086082
%Y A086079 Sequence in context: A079127 A056674 A037188 this_sequence A133703 A073265 A025438
%K A086079 base,nonn
%O A086079 0,88
%A A086079 Jason Earls (jcearls(AT)cableone.net), Jul 08 2003

%I A133703
%S A133703 1,0,1,1,0,2,0,0,0,1,1,0,0,0,2,0,1,0,0,0,2,1,0,0,0,0,0,2,0,0,0,0,0,0,0,
%T A133703 1,1,0,1,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,2
%N A133703 A054525 * A133701.
%C A133703 Right border = A001227: (1, 1, 2, 1, 2, 2, 2, 1, 3,...), the number of odd divisors of n. Row sums = A023136: (1, 1, 3, 1, 3, 3, 3, 1, 5, 3,...).
%F A133703 Mobius transform of triangle A133701.
%e A133703 First few rows of the triangle are:
%e A133703 1;
%e A133703 0, 1;
%e A133703 1, 0, 2;
%e A133703 0, 0, 0, 1;
%e A133703 1, 0, 0, 0, 2;
%e A133703 0, 1, 0, 0, 0, 2;
%e A133703 1, 0, 0, 0, 0, 0, 2;
%e A133703 ...
%Y A133703 Cf. A054525, A133701, A001227, A023136.
%Y A133703 Adjacent sequences: A133700 A133701 A133702 this_sequence A133704 A133705 A133706
%Y A133703 Sequence in context: A056674 A037188 A086079 this_sequence A073265 A025438 A030216
%K A133703 nonn,tabl
%O A133703 1,6
%A A133703 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 21 2007

%I A073265
%S A073265 1,1,0,0,1,0,1,2,0,0,0,1,1,0,0,0,2,3,0,0,0,0,2,3,1,0,0,0,1,0,4,4,0,0,0,
%T A073265 0,0,1,6,6,1,0,0,0,0,0,2,3,8,5,0,0,0,0,0,0,2,3,13,10,1,0,0,0,0,0,0,0,6,
%U A073265 12,15,6,0,0,0,0,0,0,0,2,6,10,25,15,1,0,0,0,0,0,0,0,0,4,16,31,26,7,0,0
%N A073265 Table T(n,k) (listed antidiagonalwise in order T(1,1), T(2,1), T(1,2), T(3,1), T(2,2), ...) giving the number of ordered partitions of n into exactly k powers of 2.
%D A073265 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 A073265 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).
%F A073265 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
%e A073265 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, and it is recursively computed from T(5,2)+T(4,2)+T(2,2) = 2+1+1.
%Y A073265 The first row is equal to the characteristic function of A000079, i.e. A036987 with offset 1 instead of 0, and the second row is A073267. The column sums give A023359. A073266 gives the upper triangular region of this array.
%Y A073265 Adjacent sequences: A073262 A073263 A073264 this_sequence A073266 A073267 A073268
%Y A073265 Sequence in context: A037188 A086079 A133703 this_sequence A025438 A030216 A106405
%K A073265 nonn,tabl
%O A073265 1,8
%A A073265 Antti Karttunen Jun 25 2002

%I A025438
%S A025438 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
%T A025438 0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,1,0,0,1,1,0,1,0,0,0,0,1,1,0,1,2,0,0,0,1,1,
%U A025438 0,0,1,2,0,0,3,2,0,1,1,0,1,1,1,2,1,0,3,1,0,1,3,3,0,0,1,3,1,0,3,2,0,2,2,1
%N A025438 Number of partitions of n into 5 distinct squares.
%Y A025438 Adjacent sequences: A025435 A025436 A025437 this_sequence A025439 A025440 A025441
%Y A025438 Sequence in context: A086079 A133703 A073265 this_sequence A030216 A106405 A089310
%K A025438 nonn
%O A025438 0,67
%A A025438 David W. Wilson (davidwwilson(AT)comcast.net)

%I A030216
%S A030216 1,0,0,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,0,0,1,1,0,
%T A030216 0,0,0,0,0,1,0,0,2,0,0,0,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,
%U A030216 1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,2,1,0,0,0,0,0
%V A030216 1,0,0,0,0,-1,0,-1,0,0,-1,0,1,0,-1,0,0,1,0,1,0,0,0,0,1,1,0,
%W A030216 0,0,0,0,0,-1,0,0,2,0,0,0,-1,-1,0,-1,0,0,-1,0,0,0,0,0,0,0,0,
%X A030216 -1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,2,-1,0,0,0,0,0
%N A030216 Expansion of eta(q^10)*eta(q^14).
%D A030216 M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
%Y A030216 Adjacent sequences: A030213 A030214 A030215 this_sequence A030217 A030218 A030219
%Y A030216 Sequence in context: A133703 A073265 A025438 this_sequence A106405 A089310 A129753
%K A030216 sign
%O A030216 0,36
%A A030216 njas

%I A106405
%S A106405 0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,0,1,0,0,1,0,1,
%T A106405 1,0,0,1,0,0,1,0,0,2,0,0,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0,2,0,1,1,0,0,1,1,
%U A106405 0,1,0,0,2,0,1,1,0,0,1,0,0,1,1,0,1,0,0,2,1,0,1,0,1,0,0,1,2,1,0,1,0,0,3
%N A106405 Number of odd semiprimes dividing n.
%C A106405 a(n) = A086971(n) - A106404(n);
%C A106405 a(A046315(n)) = 1; a(A093641(n)) = 0; a(A105441(n)) > 0.
%e A106405 a(105) = #{15, 21, 35} = #{3*5, 3*7, 5*7} = 3.
%Y A106405 Cf. A001227, A000005.
%Y A106405 Adjacent sequences: A106402 A106403 A106404 this_sequence A106406 A106407 A106408
%Y A106405 Sequence in context: A073265 A025438 A030216 this_sequence A089310 A129753 A070936
%K A106405 nonn
%O A106405 1,45
%A A106405 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 02 2005

%I A089310
%S A089310 0,0,0,0,0,1,0,0,0,1,1,1,0,2,0,0,0,1,1,1,1,1,1,1,0,2,2,2,0,3,0,0,0,1,1,
%T A089310 1,1,1,1,1,1,1,1,1,1,2,1,1,0,2,2,2,2,1,2,2,0,3,3,3,0,4,0,0,0,1,1,1,1,1,
%U A089310 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,3,1,1,0,2,2,2
%N A089310 Write n in binary; a(n) = number of 1's in second block of 1's from right.
%e A089310 13 = 1101 so a(13) = 2.
%o A089310 (PARI) a(n)=local(b,c,s):b=binary(n):c=length(b):while(!b[c],c=c-1):while(c>0&&b[c],c=c-1):if(c<=0,0, while(!b[c],c=c-1):s=0:while(c>