The Database of Integer Sequences, Part 8 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 ) Seis.html: Welcome index.html: Lookup indexfr.html: Francais demo1.html: Demos Sindx.html: Index WebCam.html: WebCam Submit.html: Contribute new sequence or comment eishelp1.html: Internal format eishelp2.html: Beautified format transforms.html: Transforms Spuzzle.html: Puzzles Shot.html: Hot classic.html: Classics ol.html: Superseeker JIS/index.html: Journal of Integer Sequences pages.html: More pages Maintained by: N. J. A. Sloane (njas@research.att.com), home page: www.research.att.com/~njas/ (start) %I A127249 %S A127249 1,2,1,2,2,1,0,0,0,1,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2, %T A127249 1,0,0,0,0,0,0,2,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0 %N A127249 A product of Thue-Morse related triangles. %C A127249 Product of A127243 with A127247. Inverse A127251 is given by (-1)^(n+k)T(n,k). %e A127249 Triangle begins %e A127249 1, %e A127249 2, 1, %e A127249 2, 2, 1, %e A127249 0, 0, 0, 1, %e A127249 0, 0, 0, 2, 1, %e A127249 0, 0, 0, 0, 0, 1, %e A127249 0, 0, 0, 0, 0, 0, 1, %e A127249 0, 0, 0, 0, 0, 0, 2, 1, %e A127249 0, 0, 0, 0, 0, 0, 2, 2, 1, %e A127249 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, %e A127249 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 %Y A127249 Adjacent sequences: A127246 A127247 A127248 this_sequence A127250 A127251 A127252 %Y A127249 Sequence in context: A008441 A134343 A108804 this_sequence A127251 A063251 A060208 %K A127249 easy,nonn %O A127249 0,2 %A A127249 Paul Barry (pbarry(AT)wit.ie), Jan 10 2007 %I A127251 %S A127251 1,2,1,2,2,1,0,0,0,1,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2, %T A127251 1,0,0,0,0,0,0,2,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1 %V A127251 1,-2,1,2,-2,1,0,0,0,1,0,0,0,-2,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-2,1,0,0,0,0,0, %W A127251 0,2,-2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1 %N A127251 Inverse of number triangle A127249. %C A127251 Product of A127248 and A127244. Row sums are A127252. %e A127251 Triangle begins %e A127251 1, %e A127251 -2, 1, %e A127251 2, -2, 1, %e A127251 0, 0, 0, 1, %e A127251 0, 0, 0, -2, 1, %e A127251 0, 0, 0, 0, 0, 1, %e A127251 0, 0, 0, 0, 0, 0, 1, %e A127251 0, 0, 0, 0, 0, 0, -2, 1, %e A127251 0, 0, 0, 0, 0, 0, 2, -2, 1, %e A127251 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, %e A127251 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 %Y A127251 Adjacent sequences: A127248 A127249 A127250 this_sequence A127252 A127253 A127254 %Y A127251 Sequence in context: A134343 A108804 A127249 this_sequence A063251 A060208 A004570 %K A127251 sign,tabl %O A127251 0,2 %A A127251 Paul Barry (pbarry(AT)wit.ie), Jan 10 2007 %I A063251 %S A063251 0,0,1,0,1,1,1,0,1,1,2,1,2,2,1,0,1,1,2,1,2,2,2,1,2,2,3,2,2,2,1,0,1,1,2, %T A063251 1,2,2,2,1,2,2,3,2,3,3,2,1,2,2,3,2,3,3,3,2,3,3,3,3,2,2,1,0,1,1,2,1,2,2, %U A063251 2,1,2,2,3,2,3,3,2,1,2,2,3,2,3,3,3,2,3,3,4,3,3,4,2,1,2,2,3,2,3,3,3,2,3 %N A063251 Least number of binary rotations needed to reach fixed point (with either left or right rotation allowed at each iteration). %C A063251 Fixed points are of the form 2^k-1. Left rotation is A006257, right rotation is A038572. Only-left order is A048881, only-right order is A063250. n for which mixed L/R order beats min of only-left, only-right is A063252. %e A063251 a(22)=2 with 22 right-> 11 left-> 7. (only-left requires 3, only-right requires 4) %Y A063251 A006257, A038572, A048881, A063250, A063252. %Y A063251 Adjacent sequences: A063248 A063249 A063250 this_sequence A063252 A063253 A063254 %Y A063251 Sequence in context: A108804 A127249 A127251 this_sequence A060208 A004570 A071429 %K A063251 base,easy,nonn %O A063251 0,11 %A A063251 Marc LeBrun (mlb(AT)well.com), Jul 11 2001 %I A060208 %S A060208 1,0,1,0,2,1,2,2,1,0,2,1,3,3,2,1,3,3,4,4,3,2,4,3,3,3,2,2,4,3,4,4,4,3,3,2,3,3,3,2,4,3, %T A060208 5,5,4,4,6,6,5,5,4,3,5,4,3,3,2,2,4,4,6,6,6,5,5,4,6,6,5,4,6,6,8,8,7,6,6,6,7,7,7,6,8,7, %U A060208 7,7,6,6,8,7,6,6,6,6,6,5,6,6,5,4,6,6,8,8,8,7,9,9,11,11,11,10,12,11,10,10,9,9,9,8,7,7 %V A060208 -1,0,1,0,2,1,2,2,1,0,2,1,3,3,2,1,3,3,4,4,3,2,4,3,3,3,2,2,4,3,4,4,4,3,3,2,3,3,3,2,4,3, %W A060208 5,5,4,4,6,6,5,5,4,3,5,4,3,3,2,2,4,4,6,6,6,5,5,4,6,6,5,4,6,6,8,8,7,6,6,6,7,7,7,6,8,7, %X A060208 7,7,6,6,8,7,6,6,6,6,6,5,6,6,5,4,6,6,8,8,8,7,9,9,11,11,11,10,12,11,10,10,9,9,9,8,7,7 %N A060208 2Pi(n) - Pi(2n), where Pi is A000720. %D A060208 S. Segal, On Pi(x+1)<=Pi(x)+Pi(y). Transactions American Mathematical Society, 104 (1962), 523-527. %H A060208 T. D. Noe, Table of n, a(n) for n=1..10000 %H A060208 E. Labos, Illustration %F A060208 a(n) = Mod[2*PrimePi[n], PrimePi[2n]] = 2*A000720(n)-A000720(2n) for n>1. %F A060208 a(n) ~ 2n log 2 / (log n)^2, by the prime number theorem. - njas, Mar 12 2007 %F A060208 a(n) = -A047886(n,n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 15 2008 %e A060208 n=100, Pi(100)=25, Pi(200)=46, 2*Pi(100)-Pi(2*100) =4=a(100) %Y A060208 Cf. A060207, A007097, A000720, A033844. %Y A060208 Adjacent sequences: A060205 A060206 A060207 this_sequence A060209 A060210 A060211 %Y A060208 Sequence in context: A127249 A127251 A063251 this_sequence A004570 A071429 A109672 %K A060208 sign %O A060208 1,5 %A A060208 Labos E. (labos(AT)ana.sote.hu), Mar 19 2001 %I A004570 %S A004570 2,1,2,2,1,0,2,2,0,2,0,2,2,2,0,1,1,1,1,2,0,1,0,0,0,2,2,2,0,1,2,1,0, %T A004570 1,2,0,1,2,1,2,2,0,0,0,0,0,0,2,0,1,1,1,2,1,1,0,1,0,1,2,0,2,2,1,2,2, %U A004570 0,2,0,0,0,2,0,1,1,1,0,0,2,1,2,1,1,1,1,1,1,2,1,0,1,2,2,2,2,1,2,2,1 %N A004570 Expansion of sqrt(7) in base 3. %Y A004570 Adjacent sequences: A004567 A004568 A004569 this_sequence A004571 A004572 A004573 %Y A004570 Sequence in context: A127251 A063251 A060208 this_sequence A071429 A109672 A025917 %K A004570 nonn,base,cons %O A004570 1,1 %A A004570 njas %I A071429 %S A071429 1,0,0,1,0,2,1,2,2,1,0,4,1,4,4,1,2,2,1,2,0,1,0,4,1,2,6,1,6,4,1,4,0,1,0, %T A071429 2,1,2,6,1,4,4,1,4,6,1,2,2,1,2,0,1,0,8,1,6,6,1,6,4,1,4,0,1,0,2,1,2,2,1, %U A071429 0,4,1,4,6,1,6,2,1,2,0,1,0,8,1,2,6,1,6,4,1,4,0,1,0,2,1,2,6,1,0,4 %N A071429 Sprague-Grundy values for octal game .14. %D A071429 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, p. 102 %H A071429 Achim Flammenkamp, Octal games %Y A071429 Adjacent sequences: A071426 A071427 A071428 this_sequence A071430 A071431 A071432 %Y A071429 Sequence in context: A063251 A060208 A004570 this_sequence A109672 A025917 A135689 %K A071429 nonn %O A071429 1,6 %A A071429 njas and Sue Pope (pope(AT)research.att.com), May 29 2002 %E A071429 Edited and extended by Christian G. Bower (bowerc(AT)usa.net), Oct 22 2002 %I A109672 %S A109672 1,1,1,1,1,1,1,1,2,1,2,2,1,1,1,1,3,1,1,1,1,2,2,1,2,1,1,3,3,1,3,6,3,3, %T A109672 3,1,1,2,1,1,5,5,1,2,5,2,1,1,1,1,2,5,2,1,5,5,1,1,2,1,1,3,3,3,6,3, %U A109672 1,3,3,1,1,4,6,4,1,4,12,12,4,6,12,6,4,4,1,1,3,3,1,1,7,12,7,1,3,12 %N A109672 Entries in 3-dimensional solids related to Prouhet-Tarry problem. %C A109672 Table of slices [n,k] of solids, read by antidiagonals, each slice [n,k] read by rows. %C A109672 Slice [n,0] gives A046816. %C A109672 Slice [0,k] gives A109649. %C A109672 Slice [n,n] gives A109673. %F A109672 Sum of terms in 2D slice [n, k] is 3^(n+k); example : 1+2+1+1+5+5+1+2+5+2+1+127=3^(2+1) for slice [1, 2]. %e A109672 Slice [0,0]: %e A109672 ...1... %e A109672 Slice [0,1]: %e A109672 ... 1 1 ... %e A109672 .... 1 .... %e A109672 Slice [1,0]: %e A109672 .... 1 .... %e A109672 ... 1 1... %e A109672 Slice [0,2]: %e A109672 .. 1 2 1 ... %e A109672 .... 2 2 ... %e A109672 ..... 1 ..... %e A109672 Slice [1,1]: %e A109672 ... 1 1 ... %e A109672 .. 1 3 1.. %e A109672 ... 1 1 ... %e A109672 Slice [2,0]: %e A109672 ..... 1 ..... %e A109672 .... 2 2 ... %e A109672 .. 1 2 1 ... %e A109672 Slice [0,3]: %e A109672 .. 1 3 3 1 ... %e A109672 ... 3 6 3 .... %e A109672 .... 3 3 ...... %e A109672 ..... 1 ........ %e A109672 Slice [1,2]: %e A109672 ... 1 2 1 ... %e A109672 .. 1 5 5 1 ... %e A109672 ... 2 5 2 ... %e A109672 .... 1 1 ... %e A109672 Slice [2,1]: %e A109672 .... 1 1 ... %e A109672 ... 2 5 2 ... %e A109672 .. 1 5 5 1 ... %e A109672 ... 1 2 1 ... %e A109672 Slice [3,0]: %e A109672 ..... 1 ..... %e A109672 .... 3 3 .... %e A109672 ... 3 6 3 ... %e A109672 .. 1 3 3 1 ... %Y A109672 Adjacent sequences: A109669 A109670 A109671 this_sequence A109673 A109674 A109675 %Y A109672 Sequence in context: A060208 A004570 A071429 this_sequence A025917 A135689 A029438 %K A109672 nonn,tabf,easy %O A109672 0,9 %A A109672 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 07 2005 %I A025917 %S A025917 1,0,0,0,0,0,0,1,0,0,0,1,1,0,1,0,0,0,1,1,0,1,1,1,1,1,1, %T A025917 0,1,1,1,1,1,2,1,2,2,1,1,1,2,1,2,2,2,2,2,3,2,2,2,2,2,2, %U A025917 3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5 %N A025917 Expansion of 1/((1-x^7)(1-x^11)(1-x^12)). %Y A025917 Adjacent sequences: A025914 A025915 A025916 this_sequence A025918 A025919 A025920 %Y A025917 Sequence in context: A004570 A071429 A109672 this_sequence A135689 A029438 A081592 %K A025917 nonn %O A025917 0,34 %A A025917 njas %I A135689 %S A135689 0,1,1,2,1,2,2,1,1,1,2,1,2,5,1,2,1,1,1,0,2,4,1,1,2,5,5,4,1,2,2,5,1,5,1, %T A135689 0,1,2,0,0,2,0,4,5,1,3,1,1,2,1,5,3,5,8,4,7,1,1,2,1,2,1,5,6,1,1,5,2,1,7, %U A135689 0,4,1,5,2,1,0,1,0,3,2,0,0,0,4,6,5,3,1,5,3,4,1,5,1,3,2,2,1,1,5 %V A135689 0,1,-1,-2,1,2,2,1,-1,1,-2,-1,-2,-5,-1,-2,1,1,-1,0,2,4,1,-1,2,5,5,4,1,2,2,5,-1,-5,-1,0, %W A135689 1,-2,0,0,-2,0,-4,-5,-1,-3,1,-1,-2,1,-5,-3,-5,-8,-4,-7,-1,-1,-2,-1,-2,1,-5,-6,1,1,5,2, %X A135689 1,7,0,4,-1,-5,2,1,0,-1,0,3,2,0,0,0,4,6,5,3,1,5,3,4,-1,-5,1,3,2,-2,-1,1,5 %N A135689 a(i) = if [mod[i, 2] == 0 then a(i - 2) - (a(Floor[i/2]) - a(Abs[Floor[i/2] - 1])), otherwise a[i - 1] - (a(Abs[Floor[i/2] - 2)] - a(Abs[Floor[i/2] - 3]))]. %C A135689 Recursion based on J. Mortensen's programming page for Per Norgard's "infinite series" music composition sequence technique. %H A135689 J. Mortensen, Per Norgard recursion programming %t A135689 p[0] = 0; p[1] = 1; p[2] = -1; p[3] = -2; p[i_] := p[i] = If[Mod[i, 2] == 0, p[i - 2] - (p[Floor[i/2]] - p[Abs[Floor[i/2] - 1]]), p[i - 1] - (p[Abs[Floor[i/2] - 2]] - p[Abs[Floor[i/2] - 3]])]; b = Table[p[n], {n, 0, 100}] %Y A135689 Adjacent sequences: A135686 A135687 A135688 this_sequence A135690 A135691 A135692 %Y A135689 Sequence in context: A071429 A109672 A025917 this_sequence A029438 A081592 A085028 %K A135689 nonn %O A135689 0,4 %A A135689 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 19 2008 %E A135689 Edited by njas, Mar 03 2008 %I A029438 %S A029438 1,0,0,0,0,0,0,1,0,1,1,1,0,0,1,0,1,1,2,1,2,2,1,1,1,2,1, %T A029438 3,3,3,3,3,3,2,3,3,4,4,5,5,5,5,5,5,5,6,6,7,7,8,8,8,8,8, %U A029438 9,9,10,10,11,11,12,12,12,13,13,14,14,15,15,16,17,17,18 %N A029438 Expansion of 1/((1-x^7)(1-x^9)(1-x^10)(1-x^11)). %Y A029438 Adjacent sequences: A029435 A029436 A029437 this_sequence A029439 A029440 A029441 %Y A029438 Sequence in context: A109672 A025917 A135689 this_sequence A081592 A085028 A087888 %K A029438 nonn %O A029438 0,19 %A A029438 njas %I A081592 %S A081592 1,2,1,2,2,1,1,1,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, %T A081592 1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, %U A081592 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 %N A081592 A self generating sequence: "there are n a(n)'s in the sequence". Start with 1,2 and use the rule : "a(n)=k implies there are n following k's (k is 1 or 2)". %C A081592 Lengths of runs of consecutive 1's or 2's are : 1,1,2,3,9,21,117,588 ... %e A081592 Sequence begins : 1,2 . Since a(1)=1 there is only one following "1", gives 1,2,1. Since a(2)=2 there are 2 following "2's", gives 1,2,1,2,2. Since a(3)=1 there are 3 following "1's" 1,2,1,2,2,1,1,1 etc. %Y A081592 Adjacent sequences: A081589 A081590 A081591 this_sequence A081593 A081594 A081595 %Y A081592 Sequence in context: A025917 A135689 A029438 this_sequence A085028 A087888 A109494 %K A081592 nonn %O A081592 1,2 %A A081592 Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2003 %I A085028 %S A085028 1,2,1,2,2,1,1,2,1,1,2,1,1,1,1,3,2,2,2,2,1,2,2,1,2,1,3,2,3,2,3,2,1,3,2, %T A085028 1,2,2,4,1,3,3,2,2,3,1,4,3,5,2,2,2,3,2,3,2,3,3,2,1,2,2,1,2,3,2,3,2,2,1, %U A085028 1,1,4,3,3,2,3,4,3,2,3,2,4,2,2,1,3,3,3,2,2,2,2,3,2,2,3,2,2,4 %N A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3. %C A085028 The Mobius transform of this sequence yields A057958, number of prime factors of 3^n-1. %D A085028 See references at A085021 %t A085028 Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 3]]][[2]], {n, 1, 100}] %Y A085028 Cf. A019321, A057958, A085021. %Y A085028 Adjacent sequences: A085025 A085026 A085027 this_sequence A085029 A085030 A085031 %Y A085028 Sequence in context: A135689 A029438 A081592 this_sequence A087888 A109494 A088526 %K A085028 nonn %O A085028 1,2 %A A085028 T. D. Noe (noe(AT)sspectra.com), Jun 19 2003 %I A087888 %S A087888 2,1,2,2,1,1,2,1,1,2,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1, %T A087888 1,2,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1, %U A087888 2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1 %N A087888 Given a sequence u consisting just of 1's and 2's, let f(u)(n) be the length of n-th run. Then we may define a sequence u = {a(n)} by a(n)=f^(n-1)(u)(1) (starting with n=1). %C A087888 There are exactly three infinite sequences satisfying this relation, namely this sequence, A087889 and A087890. %Y A087888 Cf. A000002, A087889, A087890. %Y A087888 Adjacent sequences: A087885 A087886 A087887 this_sequence A087889 A087890 A087891 %Y A087888 Sequence in context: A029438 A081592 A085028 this_sequence A109494 A088526 A074292 %K A087888 easy,eigen,nonn %O A087888 1,1 %A A087888 Vincent Nesme (vincent.nesme(AT)ens-lyon.fr), Oct 13 2003 %E A087888 The description was not quite clear to me but I hope I have edited it correctly. - njas %I A109494 %S A109494 1,2,1,2,2,1,1,2,1,1,5,5,1,2,5,2,1,1,1,2,1,2,8,8,1,1,8,15,8,1,2,8,8,2, %T A109494 1,2,1,1,2,1,3,11,11,3,3,18,31,18,3,1,11,31,31,11,1,2,11,18,11,2,1,3, %U A109494 3,1,1,2,1,4,14,14,4,6,32,53,32,6,4,32,80,80,32,4,1,14,53,80,53,14,1 %N A109494 Entries in 3-dimensional solid related to Prouhet-Tarry problem. %C A109494 Entries of slices [n,2] in A109672, read by rows. %C A109494 Slice [n,0] gives A046816, slice [0,k] gives A109649, slice [n,n] gives A109673, slice [n,1] gives A109390, slice [1,k] gives A109393, slice [2,k] gives A000002. %F A109494 Sum of terms in 2D slice [n, 2] is 3^(n+2). %e A109494 Slice [0,2]: %e A109494 ... 1 2 1 ... %e A109494 .... 2 2 .... %e A109494 ..... 1 ..... %e A109494 Slice [1,2]: %e A109494 ... 1 2 1 ... %e A109494 .. 1 5 5 1 ... %e A109494 ... 2 5 2 ... %e A109494 .... 1 1 .... %e A109494 Slice [2,2]: %e A109494 .... 1 2 1 .... %e A109494 ... 2 8 8 2 ... %e A109494 .. 1 8 15 8 1 ... %e A109494 ... 2 8 8 2 ... %e A109494 .... 1 2 1 .... %e A109494 Slice [3,2]: %e A109494 ..... 1 2 1 ..... %e A109494 .... 3 11 11 3 .... %e A109494 ... 3 18 31 18 3 ... %e A109494 .. 1 11 31 31 11 1 ... %e A109494 ... 2 11 18 11 2 ... %e A109494 .... 1 3 3 1 .... %e A109494 Slice [4,2]: %e A109494 ...... 1 2 1 ...... %e A109494 ..... 4 14 14 4 ..... %e A109494 .... 6 32 53 32 6 .... %e A109494 ... 4 32 80 80 32 4 ... %e A109494 .. 1 14 53 80 53 14 1 ... %e A109494 ... 2 14 32 32 14 2 ... %e A109494 .... 1 4 6 4 1 .... %e A109494 Slice [5,2]: %e A109494 ....... 1 2 1 ....... %e A109494 ...... 5 17 17 5 ...... %e A109494 ..... 10 50 81 50 10 ..... %e A109494 .... 10 70 165 165 70 10 .... %e A109494 ... 5 50 165 240 165 50 5 ... %e A109494 .. 1 17 81 165 165 81 17 1 ... %e A109494 ... 2 17 50 70 50 17 2 ... %e A109494 .... 1 5 10 10 5 1 .... %Y A109494 Cf. A000002. %Y A109494 Adjacent sequences: A109491 A109492 A109493 this_sequence A109495 A109496 A109497 %Y A109494 Sequence in context: A081592 A085028 A087888 this_sequence A088526 A074292 A097867 %K A109494 nonn,tabf,easy %O A109494 0,2 %A A109494 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 29 2005 %I A088526 %S A088526 2,1,2,2,1,1,2,1,2,1,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1 %N A088526 Duplicate of A074292. %Y A088526 Adjacent sequences: A088523 A088524 A088525 this_sequence A088527 A088528 A088529 %Y A088526 Sequence in context: A085028 A087888 A109494 this_sequence A074292 A097867 A075344 %K A088526 dead %O A088526 0,1 %I A074292 %S A074292 2,1,2,2,1,1,2,1,2,1,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1, %T A074292 2,2,1,1,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,1,1,2,1,2,1,1,2,1,2,2,1, %U A074292 2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,1,1,2,2,1,2,1,1,2,2,1,2,2,1,2 %N A074292 Dominant digit in successive groups of 3 from the Kolakoski sequence (A000002). %C A074292 This appears to be the same as a sequence studied by Claude Lenormand in a letter dated Nov 17 2003: break up the Kolakoski sequence (A000002) into runs of identical symbols, and omit one symbol from each run. %F A074292 a(n)=A000002(3n-2)+A000002(3n-1)+A000002(3n)-3. - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 15 2003 %F A074292 a(n)=A000002(A078649(n)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 16 2005 %e A074292 Kolakoski begins (1,2,2),(1,1,2),(1,2,2),(1,2,2), so this begins 2,1,2,2. %Y A074292 Adjacent sequences: A074289 A074290 A074291 this_sequence A074293 A074294 A074295 %Y A074292 Sequence in context: A087888 A109494 A088526 this_sequence A097867 A075344 A054350 %K A074292 nonn %O A074292 0,1 %A A074292 Jon Perry (perry(AT)globalnet.co.uk), Sep 21 2002 %E A074292 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 16 2003 %I A097867 %S A097867 0,1,1,1,1,2,1,2,2,1,1,2,1,2,2,1,2,3,1,2,2,1,2,3,2,3,4,1,2,3,2,3,4,2,4, %T A097867 5,2,3,4,2,4,5,3,5,7,2,4,5,3,5,7,4,7,9,3,5,7,4,7,9,5,9,12,4,7,9,5,9,12, %U A097867 7,12,16,5,9,12,7,12,16,9,16,21,7,12,16,9,16,21,12,21,28,9,16,21,12,21 %N A097867 The n-th group (n>=0) of 9 consecutive terms are the entries, read by rows, of the 3 by 3 matrix A[n]=MA[n-1], where M is the 3 by 3 matrix [[0, 1, 0], [0, 0, 1], [1, 1, 0]] and A[0] is the 3 by 3 matrix [[0, 1, 1], [1, 1, 2], [1, 2, 2]]. %e A097867 Since MA[0]=[[1,1,2],[1,2,2],[1,2,3]), the 1-st group (following the 0-th group) of 9 terms are 1,1,2,1,2,2,1,2,3. %p A097867 with(linalg): M:=matrix(3,3,[0,1,0,0,0,1,1,1,0]): A[0]:=matrix(3,3,[0,1,1,1,1,2,1,2,2]): for n from 1 to 11 do A[n]:=multiply(M,A[n-1]) od: seq(seq(seq(A[k][i,j],j=1..3),i=1..3),k=0..11); %Y A097867 Adjacent sequences: A097864 A097865 A097866 this_sequence A097868 A097869 A097870 %Y A097867 Sequence in context: A109494 A088526 A074292 this_sequence A075344 A054350 A026606 %K A097867 nonn %O A097867 0,6 %A A097867 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 31 2004 %E A097867 Edited by njas, May 20 2006 %I A075344 %S A075344 2,1,2,2,1,1,2,1,3,4,2,5,2,3,3,3,5,4,4,6,2,3,4,5,5,6,6,5,3,6,7,7,5,6,5, %T A075344 7,7,4,6,8,8,8,5,7,5,11,4,7,11,7,11,6,8,8,6,9,8,7,10,6,10,10,11,7,8,7, %U A075344 15,11,8,9,9,9,9,8,14,11,12,11,12,10,14,10,10,10,14,10,11,12,12,14,7,18 %N A075344 Number of primes between consecutive terms of A075343 (excluding endpoints). %C A075344 Conjecture: a(n) > 0 for all n. %F A075344 a(n) = A075343(n+1)-A075343(n)-n-1. - David Wasserman (dwasserm(AT)earthlink.net), Jan 16 2005 %o A075344 (PARI) i = 1; for (n = 1, 105, p = 0; for (j = 1, n, i++; while (isprime(i), p++; i++)); i++; print1(p, " ")) (Wasserman) %Y A075344 Cf. A075343. %Y A075344 Adjacent sequences: A075341 A075342 A075343 this_sequence A075345 A075346 A075347 %Y A075344 Sequence in context: A088526 A074292 A097867 this_sequence A054350 A026606 A095955 %K A075344 nonn %O A075344 1,1 %A A075344 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 18 2002 %E A075344 More terms from David Wasserman (dwasserm(AT)earthlink.net), Jan 16 2005 %I A054350 %S A054350 1,1,2,1,2,2,1,1,2,2,1,1,2,1,1,2,2,1,1,2,1,2,2,1,2,1,2,2,1,1,2,1,2,2,1, %T A054350 2,2,1,1,2,1,1,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2, %U A054350 1,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1 %N A054350 Triangular array whose rows are successive generations of the Kolakoski sequence A000002. %e A054350 1; 1,2; 1,2,2,1; 1,2,2,1,1,2,1; ... %Y A054350 Cf. A054348-A054352, A042942, A000002. %Y A054350 Row lengths give A054352. %Y A054350 Adjacent sequences: A054347 A054348 A054349 this_sequence A054351 A054352 A054353 %Y A054350 Sequence in context: A074292 A097867 A075344 this_sequence A026606 A095955 A078573 %K A054350 nonn,tabf,easy %O A054350 0,3 %A A054350 njas, May 07 2000 %E A054350 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003 %I A026606 %S A026606 1,2,1,2,2,1,1,2,2,1,1,2,2,1,1,2,1,2,1,2,2,1,1,2,2,1,1,2,1,2, %T A026606 1,2,1,2,2,1,2,1,1,2,1,2,1,2,1,2,2,1,1,2,2,1,1,2,1,2,2,1,1,2, %U A026606 2,1,1,2,1,2,1,2,1,2,2,1,2,1,1,2,1,2,1,2,1,2,2,1,1,2,2,1,1,2 %N A026606 a(n) = b(n)-1, where b(n) = n-th term of A026600 that is not a 1. %Y A026606 Adjacent sequences: A026603 A026604 A026605 this_sequence A026607 A026608 A026609 %Y A026606 Sequence in context: A097867 A075344 A054350 this_sequence A095955 A078573 A035176 %K A026606 nonn %O A026606 1,2 %A A026606 Clark Kimberling (ck6(AT)evansville.edu) %I A095955 %S A095955 1,1,1,2,1,2,2,1,1,2,2,1,2,1,1,3,2,3,1,1,3,1,1,3,3,1,3,3,1,3,3,2,3,3,3, %T A095955 2,3,3,3,3,1,2,1,3,3,3,3,2,2,2,3,2,3,2,3,2,2,3,3,2,3,2,2,2,3,2,2,2,2,2, %U A095955 3,2,2,2,2,2,2,2,2,2,2,2,3,2,2,3,2,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 %N A095955 Function f(x)=phi(sigma(x)) is iterated with initial value=n; a(n) is the length of cycle into which the trajectory merges. %C A095955 Diagnosis of true cycle of length m: a[j-m]=a[j], but a[j-d]=a[j] cases are excluded for d dividing m. %C A095955 Length 5 is rare. Example: a[6634509269055173050761216000]=5 and the 5-cycle is {6634509269055173050761216000, 7521613519844726223667200000, 7946886558074859593662464000, 7794495412499746337587200000, 7970172471593905204651622400, 6634509269055173050761216000}. The initial values 2^79=604462909807314587353088 and 2^83= 9671406556917033397649408 after more than 250 transient terms reach this cycle. %e A095955 Occurrences of cycle lengths if n<=1000: {C1=110, C2=781, C3=36, C4=67, C5=0, C6=6, C7=0...}. %t A095955 g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Jul 14 2004) %Y A095955 Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095956. %Y A095955 Adjacent sequences: A095952 A095953 A095954 this_sequence A095956 A095957 A095958 %Y A095955 Sequence in context: A075344 A054350 A026606 this_sequence A078573 A035176 A011793 %K A095955 nonn %O A095955 1,4 %A A095955 Labos E. (labos(AT)ana.sote.hu), Jul 13 2004 %I A078573 %S A078573 2,1,2,2,1,1,2,3,1,3,1,2,2,6,2,2,4,3,1,3,2,2,4,1,2,1,3,1,1,2,4,1,2,1,2, %T A078573 2,3,2,2,2,7,1,2,1,1,3,2,2,1,4,3,4,2,1,1,2,4,1,2,2,3,2,1,3,6,1,2,1,4,1, %U A078573 2,1,2,5,1,7,3,1,2,1,3,3,4,5,2,2,2,2,5,2,1,1,2,3,2,1,2,3,3,2,2,1,1,1,2 %N A078573 Maximal exponent in prime factorization of the average of n-th twin prime pair. %C A078573 a(n)=A051903(A014574(n)). %e A078573 10-th twin prime pair = (A001359(10), A006512(10)) = (107,109), hence A014574(10) = 108 = 2^2 * 3^2, therefore a(10) = 3. %Y A078573 Cf. A078570, A078572. %Y A078573 Adjacent sequences: A078570 A078571 A078572 this_sequence A078574 A078575 A078576 %Y A078573 Sequence in context: A054350 A026606 A095955 this_sequence A035176 A011793 A109649 %K A078573 nonn %O A078573 1,1 %A A078573 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 29 2002 %I A035176 %S A035176 1,1,2,1,2,2,1,1,3,2,0,2,2,1,4,1,0,3,2,2,2,0,2,2,3,2,4, %T A035176 1,0,4,0,1,0,0,2,3,0,2,4,2,0,2,0,0,6,2,0,2,1,3,0,2,0,4, %U A035176 0,1,4,0,2,4,2,0,3,1,4,0,0,0,4,2,2,3,0,0,6,2,0,4,2,2,5 %N A035176 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -14. %o A035176 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)) %Y A035176 Adjacent sequences: A035173 A035174 A035175 this_sequence A035177 A035178 A035179 %Y A035176 Sequence in context: A026606 A095955 A078573 this_sequence A011793 A109649 A098199 %K A035176 nonn %O A035176 1,3 %A A035176 njas %I A011793 %S A011793 1,1,1,1,1,1,1,1,1,2,1,2,2,1,1,3,3,1,2,5,3,1,1,5,7,4,1, %T A011793 3,8,9,4,1,1,7,14,12,5,1,3,14,20,15,5,1,1,9,25,30,18,6,1, %U A011793 4,20,42,40,22,6,1,1,12,42,66,55,26,7,1 %N A011793 Triangle of numbers of irreducible Euler sums. %H A011793 D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory %Y A011793 Adjacent sequences: A011790 A011791 A011792 this_sequence A011794 A011795 A011796 %Y A011793 Sequence in context: A095955 A078573 A035176 this_sequence A109649 A098199 A022828 %K A011793 nonn,tabl,nice %O A011793 1,10 %A A011793 njas, David Broadhurst (D.Broadhurst(AT)open.ac.uk) %I A109649 %S A109649 1,1,1,1,2,1,2,2,1,1,3,3,1,3,6,3,3,3,1,1,4,6,4,1,4,12,12,4,6,12,6,4,4, %T A109649 1,1,5,10,10,5,1,5,20,30,20,5,10,30,30,10,10,20,10,5,5,1,1,6,15, %U A109649 20,15,6,1,6,30,60,60,30,6,15,60,90,60,15,20,60,60,20,15,30,15,6 %N A109649 Entries in 3-dimensional version of Pascal triangle: trinomial coefficients. %C A109649 Greatest numbers in each 2D triangle form A022916 = (multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!). %C A109649 2D triangle sums are powers of 3. %C A109649 See A046816 for another version. %F A109649 Coefficients of x, y, z in (x+y+z)^n. %e A109649 .1 3 3 1 ... Here is the third slice of the pyramid %e A109649 . 3 6 3 %e A109649 .. 3 3 %e A109649 ... 1 ..... %Y A109649 Cf. A007318, A046816, A002378, A027480, A033487, A033488. %Y A109649 Adjacent sequences: A109646 A109647 A109648 this_sequence A109650 A109651 A109652 %Y A109649 Sequence in context: A078573 A035176 A011793 this_sequence A098199 A022828 A129406 %K A109649 nonn,tabf,easy %O A109649 0,5 %A A109649 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 03 2005 %I A098199 %S A098199 2,1,2,2,1,1,46,459,2,3,1,2,2,1,1,8,18,1,1,1,1,6,5,6,14,1,2,1,1,140,1,2, %T A098199 1,1,2,1,9,16,1,2,1,1,1,15,1,3,55,1,1,12,1,1,5,4,6,13,2,2,7,2,32,1,1,6, %U A098199 1,1,54,1,1,1,21,1,2,1,3,4,5,15,1,6,1,2,5,1,1,7,1,834,2,1,4,8,3,2,3,1,5 %N A098199 List of the partial quotients of the continued fraction for (Pi^4)/36~2.705808084277845, the limiting value of ratio A024916[n]/A002088[n]=SummatorySigma/SummatoryTotient. %t A098199 <Fibonacci Polynomial %F A123018 T(n, k) := Binomial[n - k - 1, k] (* Bezier transform:*) T'(n,k)=T(n, k)*x^k*(1 - x)^(n - k) %e A123018 Triangle begins: %e A123018 1 %e A123018 1,-2 %e A123018 1, -2, 2 %e A123018 1,-2, 1, -1 %e A123018 1, -2, 0, 2 %e A123018 1, -2,-1, 5,-4 %e A123018 1, -2, -2, 8,-7, 2, 1 %t A123018 T[n_, k_] := Binomial[n - k - 1, k]; a = Table[CoefficientList[Sum[T[n, k]*x^k*(1 - x)^(n - k), {k, 0, n}], x], {n, 0, 10}]; Flatten[a] %Y A123018 Cf. A049310. %Y A123018 Adjacent sequences: A123015 A123016 A123017 this_sequence A123019 A123020 A123021 %Y A123018 Sequence in context: A098199 A022828 A129406 this_sequence A100429 A049710 A025143 %K A123018 sign,tabf,easy,more %O A123018 0,3 %A A123018 Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 24 2006 %E A123018 Edited by njas, May 26 2007 %I A100429 %S A100429 2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,2,1,1, %T A100429 2,2,1,1,2,1,2,1,1,2,2,1,2,1,2,2,1,1,2,2,1,2,1,1,2,1,1,2,1,1,2,1,2,2,1, %U A100429 2,2,1,1,2,1,2,2,1,1,2,1,2,1,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,2,1,2,2,1,2 %N A100429 Bisection of Kolakoski sequence A000002. %Y A100429 Adjacent sequences: A100426 A100427 A100428 this_sequence A100430 A100431 A100432 %Y A100429 Sequence in context: A022828 A129406 A123018 this_sequence A049710 A025143 A080634 %K A100429 nonn,easy %O A100429 0,1 %A A100429 njas, Nov 20 2004 %E A100429 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 12 2006 %I A049710 %S A049710 2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,2,1,1, %T A049710 2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2, %U A049710 1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2 %N A049710 a(n)=3-k(n), where k=A006928; also, a and k have the same runlength sequence, with n-th term k(n-1) for n >= 2. %Y A049710 Adjacent sequences: A049707 A049708 A049709 this_sequence A049711 A049712 A049713 %Y A049710 Sequence in context: A129406 A123018 A100429 this_sequence A025143 A080634 A109925 %K A049710 nonn %O A049710 1,1 %A A049710 Clark Kimberling (ck6(AT)evansville.edu) %I A025143 %S A025143 2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2, %T A025143 2,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1, %U A025143 2,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2 %N A025143 Unique sequence a such that a(1) = 2 and r(r(a)) = a != r(a), where for any sequence s of 1's and 2's, r(s(n) := length of n-th run of same symbols in s; r(a) is sequence A025142. %D A025143 C. Kimberling, Problem 90: Run-length sequences, Mathematische Semesterberichte, 44 (1997) 94-95. %Y A025143 Differs from A014675 in many entries starting at entry 8. %Y A025143 Adjacent sequences: A025140 A025141 A025142 this_sequence A025144 A025145 A025146 %Y A025143 Sequence in context: A123018 A100429 A049710 this_sequence A080634 A109925 A001468 %K A025143 nonn %O A025143 1,1 %A A025143 Clark Kimberling (ck6(AT)evansville.edu) %I A080634 %S A080634 1,2,1,2,2,1,2,1,2,1,1,2,1,2,1,2,1,2,2,1,2,2,1,1,2,1,2,1,2,2,2,1,1,1,2, %T A080634 2,1,2,1,1,2,2,2,1,1,1,2,1,1,2,1,2,1,2,1,2,1,1,2,2,2,2,2,1,2,1,1,1,2,1, %U A080634 2,2,2,1,2,2,1,2,1,2,1,2,1,1,2,1,2,2,2,1,2,1,1,2,1,2,1,1,2,2,2,1,2,1,2 %N A080634 Start with a(1)=1. Then, for n>1, choose a(n)=1 or 2 so as to minimize the longest arithmetic progression in either S1(n) or S2(n), where S1(n)={k|a(k)=1,1<=k<=n} and S2(n)={k|a(k)=2,1<=k<=n}. %e A080634 Given the first seven terms as {1,2,1,2,2,1,2}, we find that a(8)=1 gives S1(8)={1,3,6,8}, S2(8)={2,4,5,7}, both with maximum AP's of length 2, whereas a(8)=2 gives S1(8)={1,3,6}, S2(8)={2,4,5,7,8}, with max length AP's of 2 for S1 and 3 for S2. So a(8) must be assigned the value of 1. %Y A080634 Adjacent sequences: A080631 A080632 A080633 this_sequence A080635 A080636 A080637 %Y A080634 Sequence in context: A100429 A049710 A025143 this_sequence A109925 A001468 A014675 %K A080634 nonn %O A080634 1,2 %A A080634 John W. Layman (layman(AT)math.vt.edu), Feb 27 2003 %I A109925 %S A109925 0,0,1,2,1,2,2,1,2,1,2,1,2,1,3,0,1,2,3,1,4,0,2,1,2,0,3,0,1,1,2,1,3,1,3, %T A109925 0,2,1,4,0,1,1,2,1,5,0,2,1,3,0,3,0,1,1,3,0,2,0,1,1,3,1,4,0,1,1,2,1,5,0, %U A109925 2,1,2,1,6,0,3,0,2,1,3,0,3,1,2,0,4,0,1,1,3,0,3,0,2,0,1,1,3,0,2,1,2,1,6 %N A109925 Number of primes of the form n - 2^k. %C A109925 Erdos conjectures that the numbers in A039669 are the only n for which n-2^r is prime for all 2^r0; A118952(n)<=a(n); A078687(n)=a(A000040(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 07 2006 %e A109925 a(21) = 4, 21-2 =19, 21-4 = 17, 21-8 = 13, 21-16 = 5, four primes. %t A109925 Table[cnt=0; r=1; While[r12, 2->122, take limit . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 23 2005 %D A001468 M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151. %D A001468 D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43. %D A001468 D. R. Hofstadter, personal communication. %D A001468 Problem E1226, Amer. Math. Monthly, 64 (1957), 197-198. %D A001468 Problem 4247, Amer. Math. Monthly, 55 (1948), 588-592. %F A001468 [(n+1) tau] - [n tau], tau =(1 + sqrt 5)/2 = A001622, [] = floor function. %p A001468 Digits := 50: t := evalf( (1+sqrt(5))/2); A001468 := n->floor((n+1)*t)-floor(n*t); %t A001468 Table[Floor[GoldenRatio*(n + 1)] - Floor[GoldenRatio*n], {n, 0, 80}] - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 %Y A001468 Same as A014675 if initial 1 is deleted. Cf. A003849. %Y A001468 Adjacent sequences: A001465 A001466 A001467 this_sequence A001469 A001470 A001471 %Y A001468 Sequence in context: A025143 A080634 A109925 this_sequence A014675 A107362 A022303 %K A001468 nonn,easy,nice %O A001468 0,2 %A A001468 njas. Rechecked Nov 07, 2001 %I A014675 %S A014675 2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1, %T A014675 2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1, %U A014675 2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2 %N A014675 The infinite Fibonacci word (start with 1, apply 1->2, 2->21, take limit). %D A014675 M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151. %D A014675 J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141. %D A014675 G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36. %D A014675 G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149. %F A014675 Define strings S(0)=1, S(1)=2, S(n)=S(n-1)S(n-2); iterate. Sequence is S(infinity). %F A014675 a(n) = [(n+1)*phi] - [n*phi], phi =(1+ sqrt 5)/2. %p A014675 Digits := 50: t := evalf( (1+sqrt(5))/2); A014675 := n->floor((n+1)*t)-floor(n*t); %t A014675 Nest[ Flatten[ # /. {1 -> 2, 2 -> {2, 1}}] &, {1}, 11] (* Robert G. Wilson v *) %Y A014675 This is the 1, 2 version. The standard form is A003849. See also A005614. First differences of A000201. %Y A014675 Cf. A082389. %Y A014675 Differs from A025143 in many entries starting at entry 8. Same as A001468 if an initial 1 is added. %Y A014675 Cf. A008351. %Y A014675 Adjacent sequences: A014672 A014673 A014674 this_sequence A014676 A014677 A014678 %Y A014675 Sequence in context: A080634 A109925 A001468 this_sequence A107362 A022303 A113189 %K A014675 nonn,easy,nice %O A014675 0,1 %A A014675 njas %E A014675 Corrected by njas, Nov 07, 2001 %I A107362 %S A107362 1,2,1,2,2,1,2,1,2,2,2,1,2,1,2,2,2,1,1,2,2,1,2,2,2,2,2,1,1,1,2,2,2,1,1, %T A107362 2,2,2,2,2,2,2,2,1,1,1,1,1,2,2,2,2,2,1,1,2,1,2,2,2,2,2,2,2,2,2,2,2,1,2, %U A107362 1,1,1,1,1,2,1,1,2,2,2,2,2,1,2,1,2,1,2,1,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2 %N A107362 Sequence A={a(n),n=0,1,2,3,...} such that the subsequence S1={a(n)|n mod 5=0,3} is identical to A, and S2=S\S1 (the complement of S1 in A) is identical to A except with the first term omitted. %F A107362 a(0)=1, a(1)=2, a(2)=1, and, for n>2, a(n)=a(2[n/5] if n=0 mod 5, a(n)=a(n-2[n/5]) if n=1, 2 mod 5, a(n)=a(2[n/5]+1) if n=3 mod 5, and a(n)=a(n-2[n/5]-1) if n=4 mod 5. %e A107362 S1={a(0),a{3),a(5),a(8),a(10),...}={1,2,1,2,2,...}=A. Similarly for A\S1. %Y A107362 Adjacent sequences: A107359 A107360 A107361 this_sequence A107363 A107364 A107365 %Y A107362 Sequence in context: A109925 A001468 A014675 this_sequence A022303 A113189 A114284 %K A107362 eigen,nonn %O A107362 0,2 %A A107362 John W. Layman (layman(AT)math.vt.edu), May 24 2005 %I A022303 %S A022303 1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1, %T A022303 2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1, %U A022303 1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1 %N A022303 a(n) = length of (n+2)nd run, with initial terms 1, 2. %Y A022303 Adjacent sequences: A022300 A022301 A022302 this_sequence A022304 A022305 A022306 %Y A022303 Sequence in context: A001468 A014675 A107362 this_sequence A113189 A114284 A023575 %K A022303 nonn %O A022303 0,2 %A A022303 Clark Kimberling (ck6(AT)evansville.edu) %I A113189 %S A113189 1,2,1,2,2,1,2,2,1,2,2,0,2,1,2,3,1,1,2,0,3,3,1,0,1,1,1,0,0,2,2,0,2,5,1, %T A113189 0,2,0,1,0,2,0,2,0,2,2,3,1,1,2,2,0,1,1,2,2,3,7,0,0,1,0,0,1,0,1,2,2,2,1, %U A113189 2,1,2,1,1,0,1,0,4,0,2,6,2,2,3,0,2,1,1,0,3,0,0,6,1,0,2,5,2,0,0,1,4,2,2 %N A113189 Number of times that Fibonacci(n)-Fibonacci(i) is prime for i=0..n-3. %C A113189 We exclude i=n-2 and i=n-1 because they yield Fibonacci(n-2) and Fibonacci(n-1), respectively. Sequence A113190 lists the n for which a(n)=0. %t A113189 Table[cnt=0; Do[If[PrimeQ[Fibonacci[n]-Fibonacci[i]], cnt++ ], {i, 0, n-3}]; cnt, {n, 3, 150}] %Y A113189 Cf. A113188 (primes that are the difference of two Fibonacci numbers). %Y A113189 Adjacent sequences: A113186 A113187 A113188 this_sequence A113190 A113191 A113192 %Y A113189 Sequence in context: A014675 A107362 A022303 this_sequence A114284 A023575 A105446 %K A113189 nonn %O A113189 3,2 %A A113189 T. D. Noe (noe(AT)sspectra.com), Oct 17 2005 %I A114284 %S A114284 1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,2,2,2, %T A114284 1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,1 %V A114284 1,-2,1,-2,-2,1,-2,-2,-2,1,-2,-2,-2,-2,1,-2,-2,-2,-2,-2,1,-2,-2,-2,-2,-2,-2,1,-2,-2,-2, %W A114284 -2,-2,-2,-2,1,-2,-2,-2,-2,-2,-2,-2,-2,1,-2,-2,-2,-2,-2,-2,-2,-2,-2,1,-2,-2,-2,-2,-2, %X A114284 -2,-2,-2,-2,-2,1 %N A114284 Riordan array ((1-3x)/(1-x),x). %C A114284 Inverse of A114283. Row sums are 1-2n. Diagonal sums are A114285. Sequence array of 3*0^n-2. %F A114284 T(n, k)=if(k<=n, 3*0^(n-k)-2, 0) %e A114284 Triangle begins %e A114284 1; %e A114284 -2,1; %e A114284 -2,-2,1; %e A114284 -2,-2,-2,1; %e A114284 -2,-2,-2,-2,1; %Y A114284 Adjacent sequences: A114281 A114282 A114283 this_sequence A114285 A114286 A114287 %Y A114284 Sequence in context: A107362 A022303 A113189 this_sequence A023575 A105446 A058978 %K A114284 easy,sign,tabl %O A114284 0,2 %A A114284 Paul Barry (pbarry(AT)wit.ie), Nov 20 2005 %I A023575 %S A023575 1,2,1,2,2,1,2,2,2,1,2,2,2,2,2,2,2,1,3,2,2,2,2,2,2,2,2,3,2,2,3,2,3, %T A023575 2,2,3,2,2,3,2,3,2,2,2,2,2,2,2,3,2,2,2,2,2,3,3,2,2,3,2,3,2,3,2,2,2, %U A023575 2,3,3,2,2,2,3,2,2,2,2,2,2,2,2,2,3,2,3,2,2,3,2,2,3,2,3,3,2,3,1,2,2 %N A023575 Number of distinct prime divisors of prime(n)+3. %Y A023575 Adjacent sequences: A023572 A023573 A023574 this_sequence A023576 A023577 A023578 %Y A023575 Sequence in context: A022303 A113189 A114284 this_sequence A105446 A058978 A118916 %K A023575 nonn %O A023575 1,2 %A A023575 Clark Kimberling (ck6(AT)evansville.edu) %I A105446 %S A105446 1,1,1,2,1,2,2,1,2,2,2,2,1,2,2,2,3,2,2,2,1,2,2,2,3,2,3,3,2,3,2,2,2,1,2, %T A105446 2,2,3,2,3,3,2,3,3,3,3,2,3,3,2,3,2,2,2,1,2,2,2,3,2,3,3,2,3,3,3,3,2,3,3, %U A105446 3,4,3,3,3,2,3,3,3,4 %N A105446 Number of symbols in the Roman Fibonacci number representation of n. %C A105446 The Roman Fibonacci numbers are composed from the values of the Fibonacci Numbers (A000045) with the grammar of the Roman Numerals (A006968) and a few rules to disambiguate. %C A105446 The alphabet: {1, 2, 3, 5, 8, A=13, B=21, C=34, D=55, E=89, F=144, ...}. Rule one: of the infinite set of representations of integers by this grammar, always restrict to the subset of those with shortest length. Rule two: if there are two or more in the subset of shortest representations, restrict to the subset with fewest subtractions [A31 preferred to 188, B31 preferred to 1AA, CA preferred to 8D, DB preferred to AE]. %C A105446 Rule three: if there are two or more representations per Rules one and two, restrict to the subset with the most duplications of characters [22 preferred to 31, 33 preferred to 51, 55 preferred to 82, 88 preferred to A3, BBB preferred to D53, CC preferred to BE]. We do not need a Rule four for a while... %C A105446 Lemma: no Roman Fibonacci number requires three consecutive instances of the same symbol. Proof: 3*F(i) = F(i+2) + F(i-2). Question: what is the asymptotic length of the Roman Fibonacci numbers? %D A105446 Cajori, F. A History of Mathematical Notations, 2 vols. Bound as One, Vol. 1: Notations in Elementary Mathematics. New York: Dover, pp. 30-37, 1993. %D A105446 Menninger, K. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover, pp. 44-45 and 281, 1992. %D A105446 Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 4-5, 1969. %H A105446 Eric Weisstein's World of Mathematics, Roman Numerals. %H A105446 Eric Weisstein's World of Mathematics, Fibonacci Numbers. %F A105446 a(n) = number of symbols in the Roman Fibonacci number representation of n, as defined in "Comments." a(n) = 1 iff n is an element of A000045. a(n) = 2 iff the shortest Roman Fibonacci number representation of n is as the sum or difference of two elements of A000045, and n is not an element of A000045. %e A105446 a(1) = 1 because 1 is a Fibonacci number, equal to its own representation as a Roman Fibonacci number. %e A105446 a(4) = 2 because 4 is not a Fibonacci number, but can be represented as the sum or difference of two Fibonacci numbers, with its Roman Fibonacci number representation being "22" (not "31" per rule three). %e A105446 a(17) = 3 because the Roman Fibonacci number representation of 17 has three symbols, namely "A22" (not "188" per rule two). %e A105446 a(80) = 4 because the Roman Fibonacci number representation of 80 has four symbols, namely "DB22". %Y A105446 A105447 = integers with A105446(n) = 2. A105448 = integers with A105446(n) = 3. A105449 = integers with A105446(n) = 4. A105450 = integers with A105446(n) = 5. A023150 = integers with A105446(n) = 6. A105452 = integers with A105446(n) = 7. A105453 = integers with A105446(n) = 8. A105454 = integers with A105446(n) = 9. A105455 = integers with A105446(n) = 10. %Y A105446 Cf. A000045, A006968. %Y A105446 Adjacent sequences: A105443 A105444 A105445 this_sequence A105447 A105448 A105449 %Y A105446 Sequence in context: A113189 A114284 A023575 this_sequence A058978 A118916 A107800 %K A105446 base,easy,nonn %O A105446 1,4 %A A105446 Jonathan Vos Post (jvospost2(AT)yahoo.com), Apr 09 2005 %I A058978 %S A058978 1,1,1,2,1,2,2,1,2,2,2,2,1,2,2,2,3,2,2,2,1,2,2,2,3,2,3,3,2,3,2,2,2,1,2, %T A058978 2,2,3,2,3,3,2,3,3,3,3,2,3,3,3,3,2,2,2,1,2,2,2,3,2,3,3,2,3,3,3,3,2,3,3, %U A058978 3,4,3,3,3,2 %N A058978 Minimal number of (non-consecutive) Fibonacci numbers needed to get n by addition and subtraction. %Y A058978 Cf. A007895. %Y A058978 Adjacent sequences: A058975 A058976 A058977 this_sequence A058979 A058980 A058981 %Y A058978 Sequence in context: A114284 A023575 A105446 this_sequence A118916 A107800 A085761 %K A058978 nonn %O A058978 1,4 %A A058978 Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 15 2001 %I A118916 %S A118916 1,2,1,2,2,1,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2, %T A118916 2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, %U A118916 2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 %N A118916 Number of inequivalent primes in ring of integers Z[sqrt(2)] of successive norms (indexed by A055029). %Y A118916 Cf. A055029, A118917. %Y A118916 Adjacent sequences: A118913 A118914 A118915 this_sequence A118917 A118918 A118919 %Y A118916 Sequence in context: A023575 A105446 A058978 this_sequence A107800 A085761 A102382 %K A118916 easy,nice,nonn %O A118916 1,2 %A A118916 Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 05 2006 %I A107800 %S A107800 1,1,1,1,1,2,1,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, %T A107800 2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, %U A107800 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,1,2,2,2,2,3,2,2,2,2,2,2 %N A107800 a(n) = number of distinct primes dividing A006049(n) (and dividing (A006049(n)+1). %C A107800 a(n) first equals 3 when n is such that A006049(n) = 230. %H A107800 T. D. Noe, Table of n, a(n) for n=1..2500 %F A107800 a(n) = A001221(A006049(n)). %t A107800 f[n_] := Length@FactorInteger[n];t = f /@ Range[300];f /@ Flatten@Position[Rest[t] - Most[t], 0] (*Chandler*) %Y A107800 Cf. A001221, A006049, A052215. %Y A107800 Adjacent sequences: A107797 A107798 A107799 this_sequence A107801 A107802 A107803 %Y A107800 Sequence in context: A105446 A058978 A118916 this_sequence A085761 A102382 A024890 %K A107800 nonn %O A107800 1,6 %A A107800 Leroy Quet (qq-quet(AT)mindspring.com), Mar 24 2007 %E A107800 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Mar 27 2007 %I A085761 %S A085761 1,1,2,1,2,2,1,2,2,2,2,2,2,3,3,2,2,3,3,3,3,2,3,3,3,3,3,4,3,3,3,3,4,4,4, %T A085761 4,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,4,4,4,4,5,5,5,5,5,5,5,4,5,5,5, %U A085761 5,5,5,5,5,5,6,6,5,5,5,5,5,5,5,6,6,6,6,6,6,5,5,5,6,6,6,6,6,6,6,6,6,6,7 %N A085761 Number of triangular numbers between n and 2n (inclusive). %Y A085761 Adjacent sequences: A085758 A085759 A085760 this_sequence A085762 A085763 A085764 %Y A085761 Sequence in context: A058978 A118916 A107800 this_sequence A102382 A024890 A007895 %K A085761 easy,nonn %O A085761 1,3 %A A085761 Amarnath Murthy, Jason Earls (jcearls(AT)cableone.net), Jul 22 2003 %I A102382 %S A102382 0,1,1,1,1,1,2,1,2,2,1,2,2,2,2,3,2,1,4,3,4,4,4,3,1,4,4,5,2,2,2,3,3,3,2, %T A102382 2,3,1,3,5,4,4,3,4,4,1,4,2,4,4,3,3,3,4,2,1,3,3,4,6,5,2,4,4,4,2,3,5,5,3, %U A102382 1,6,4,3,4,6,3,1,4,4,4,5,4,4,3,2,2,3,3,3,6,3,4,4,3,4,4,4,3,8,3,4,5,3,4 %N A102382 A001221(A004094(n)). %e A102382 n=19: 2^19=524288 -> 882425=5*5*47*751: a(19)=3. %Y A102382 Cf. A102383, A102384, A102385. %Y A102382 Adjacent sequences: A102379 A102380 A102381 this_sequence A102383 A102384 A102385 %Y A102382 Sequence in context: A118916 A107800 A085761 this_sequence A024890 A007895 A136655 %K A102382 nonn %O A102382 0,7 %A A102382 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 06 2005 %I A024890 %S A024890 0,0,1,0,1,1,1,1,2,1,2,2,1,2,2,2,2,3,3,2,3,3,2,3,3,3,3,4,4,4,4,3,4,4,3,4,4, %T A024890 3,5,5,5,4,5,5,5,5,4,5,5,5,5,5,5,6,6,5,6,6,5,5,6,6,5,6,6,6,6,5,7,7,7,7,7,6, %U A024890 7,7,7,7,7,7,6,7,7,6,7,7,7,8,7,8,7,8,8,8,8,7,8,7,8,8,8,8,7,8,8,8,8,9,8,8,9 %N A024890 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = A023531, t = A014306. %Y A024890 Adjacent sequences: A024887 A024888 A024889 this_sequence A024891 A024892 A024893 %Y A024890 Sequence in context: A107800 A085761 A102382 this_sequence A007895 A136655 A053260 %K A024890 nonn %O A024890 2,9 %A A024890 Clark Kimberling (ck6(AT)evansville.edu) %I A007895 %S A007895 0,1,1,1,2,1,2,2,1,2,2,2,3,1,2,2,2,3,2,3,3,1,2,2,2,3,2,3,3,2,3,3,3,4, %T A007895 1,2,2,2,3,2,3,3,2,3,3,3,4,2,3,3,3,4,3,4,4,1,2,2,2,3,2,3,3,2,3,3,3, %U A007895 4,2,3,3,3,4,3,4,4,2,3,3,3,4,3,4,4,3,4,4,4,5,1 %N A007895 Number of terms in Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers). %C A007895 Let M(0)=0, M(1)=1, and for i > 0, M(i+1)=f(concatenation of M(j), j from 0 to i-1) where f is the morphism f(k)=k+1. Then sequence = concatenation of M(j) for j from 0 to infinity. - Claude Lenormand (claude.lenormand(AT)free.fr), Dec 16 2003 %C A007895 a(n) = A000120(A003714(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2005 %D A007895 D. E. Daykin, Representation of natural numbers as sums of generalized Fibonacci numbers, J. London Math. Soc. 35 (1960) 143-160. %D A007895 C. G. Lekkerkerker, Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci, Simon Stevin 29 (1952) 190-195. %D A007895 F. Weinstein, The Fibonacci Partitions, preprint, 1995. %D A007895 E. Zeckendorf, Representation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liege 41, 179-182, 1972. %H A007895 T. D. Noe, Table of n, a(n) for n=0..10000 %H A007895 Joerg Arndt, Fxtbook %H A007895 I. Nemes, Fibonacci representations of multiples of Fibonacci numbers %H A007895 F. V. Weinstein, Notes on Fibonacci partitions %F A007895 a(n) = A107015(n) + A107016(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 09 2005 %e A007895 a(46) = a(1+3+8+34) = 4. %Y A007895 Cf. Cf. A035514, A035515, A035516, A035517. %Y A007895 Record positions are in A027941. %Y A007895 Adjacent sequences: A007892 A007893 A007894 this_sequence A007896 A007897 A007898 %Y A007895 Sequence in context: A085761 A102382 A024890 this_sequence A136655 A053260 A014643 %K A007895 nonn %O A007895 0,5 %A A007895 Felix Weinstein (wain(AT)ana.unibe.ch), Clark Kimberling (ck6(AT)evansville.edu) %I A136655 %S A136655 0,1,1,1,2,1,2,2,1,2,2,2,3,1,2,2,2,3,2,3,3,1,2,2,2,3,2,3,3,2,3,3,3,4,1, %T A136655 2,2,2,3,2,3,3,2,3,3,3,4,2,3,3,3,4,3,4,4,1,2,2,2,3,2,3,3,2,3,3,3,4,2,3, %U A136655 3,3,4,3,4,4,2,3,3,3,4,3,4,4,3,4,4,4,5,1,2,2,2,3,2,3,3,2,3,3,3,4,2,3,3 %N A136655 Number of 0's (or B's) in the Wythoff representation of n. %C A136655 See A135817 for references and links for the Wythoff representation for n>=1. %C A136655 Is this the same as A007895? - R. J. Mathar (mathar(AT)strw.leidenuniv.nl) Apr 29 2008 %F A136655 a(n)= number of applications of Wythoff's B sequence A001950 needed in the unique Wythoff representation of n>=1. %e A136655 16=A(B(A(A(B(1))))) = ABAAB = `10110`, hence a(16)=2. %Y A136655 Cf. A135817 (lengths of Wythoff representation). A135818 (number of 1's (or A's) in the Wythoff representation). %Y A136655 Adjacent sequences: A136652 A136653 A136654 this_sequence A136656 A136657 A136658 %Y A136655 Sequence in context: A102382 A024890 A007895 this_sequence A053260 A014643 A118382 %K A136655 nonn,easy,new %O A136655 1,5 %A A136655 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008 %I A053260 %S A053260 0,1,0,1,1,0,1,1,1,1,1,1,1,2,1,2,2,1,2,2,2,3,3,2,3,3,3,3,4,4,4,5,4,5, %T A053260 5,5,6,6,6,7,7,7,8,9,8,9,10,9,11,11,11,12,13,13,14,15,15,16,17,17,18, %U A053260 19,19,21,22,22,24,25,25,27,28,29,30,32,32,34,36,36,39,40,41,44,45,46 %N A053260 Coefficients of the '5th order' mock theta function psi_0(q) %D A053260 George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134 %D A053260 George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255 %D A053260 Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355 %D A053260 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22 %D A053260 George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304 %F A053260 G.f.: psi_0(q) = sum for n >= 0 of q^((n+1)(n+2)/2) (1+q)(1+q^2)...(1+q^n) %F A053260 a(n) = number of partitions of n such that each part occurs at most twice, the largest part is unique, and if k occurs as a part then all smaller positive integers occur %t A053260 Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}], {n, 0, 12}], {q, 0, 100}] %Y A053260 Other '5th order' mock theta functions are at A053256, A053257, A053258, A053259, A053261, A053262, A053263, A053264, A053265, A053266, A053267. %Y A053260 Adjacent sequences: A053257 A053258 A053259 this_sequence A053261 A053262 A053263 %Y A053260 Sequence in context: A024890 A007895 A136655 this_sequence A014643 A118382 A007723 %K A053260 nonn,easy %O A053260 0,14 %A A053260 Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 19 1999 %I A014643 %S A014643 1,1,1,2,1,2,2,1,2,2,3,3,1,2,2,3,3,4,4,4,5,5,5, %T A014643 1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,9,10,10,10,10,10, %U A014643 11,11,11,11,11 %N A014643 Triangular array starting with {1,1}; then i-th term in a row gives number of i's in next row. %e A014643 {1,1},{1,2},{1,2,2},{1,2,2,3,3},... %Y A014643 Cf. A014644, A011784. Rows converge to A001462. %Y A014643 Adjacent sequences: A014640 A014641 A014642 this_sequence A014644 A014645 A014646 %Y A014643 Sequence in context: A007895 A136655 A053260 this_sequence A118382 A007723 A067437 %K A014643 tabl,nonn %O A014643 1,4 %A A014643 njas %E A014643 More terms from Patrick De Geest (pdg(AT)worldofnumbers.com) %I A118382 %S A118382 1,1,2,1,2,2,1,2,3,1,1,2,3,3,1,2,1,2,4,1,1,1,1,3,2,2,3,1,2,3,4,3,2,1,1, %T A118382 1,1,2,4,1,4,2,1,1,1,3,1,2,1,5,2,2,1,2,3,1,3,3,2,6,1,2,2,1,3,1,3,2,5,1, %U A118382 1,1,1,2,2,1,4,1,4,1,4,1,4,1,2,3,3,1,2,3,3,3,3,3,1,2,1,2,1,1,2,5,1,2,2 %N A118382 Primitive Orloj clock sequences; row n sums to 2n-1. %C A118382 An Orloj clock sequence is a finite sequence of positive integers that, when iterated, can be grouped so that the groups sum to successive natural numbers. There is one primitive sequence whose values sum to each odd m; all other sequences can be obtained by repeating and refining these. Refining means splitting one or more terms into values summing to that term. The Orloj clock sequence is the one summing to 15: 1,2,3,4,3,2, with a beautiful up and down pattern. %F A118382 Let b(i),0<=i= n, tri -= n); found[tri] = 1); last = 0; r = []; for(i = 1, n, if(found[i], r = concat(r, [i-last]); last = i)); r} %Y A118382 Cf. A028355, A118383. Length of row n is A117484(2n-1) = A000224(2n-1). %Y A118382 Adjacent sequences: A118379 A118380 A118381 this_sequence A118383 A118384 A118385 %Y A118382 Sequence in context: A136655 A053260 A014643 this_sequence A007723 A067437 A029315 %K A118382 nonn,tabf %O A118382 1,3 %A A118382 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Apr 26 2006 %I A007723 %S A007723 1,1,2,1,2,2,1,2,3,2,1,2,4,4,2,1,2,5,8,5,2,1,2,6,15,16,6,2,1,2,7,26,52, %T A007723 32,7,2,1,2,8,42,152,203,64,8,2,1,2,9,64,392,1144,877,128,9,2,1,2,10,93, %U A007723 904,5345,10742,4140,256,10,2,1,2,11,130,1899,20926,102050,122772,21147 %N A007723 Triangle a(n,k) of number of M-sequences read by antidiagonals. %D A007723 S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266. %H A007723 S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266. %F A007723 a(p, n)=sum_{-1<=k<=n} Lp(n, k) where Lp(n, k) satisfies the recurrence: Lp(n, k) = sum_{k<=i<=n} L(p-1, n, i)*L(p, i-1, k-1) for p, n>=1, k>=0 with the boundary conditions: Lp(n, n)=Lp(n, -1)=1 for all p>=1, n>=-1; L0(n, n)=L0(n, -1)=1 and L0(n, k)=0 for k different from -1 or n - Pab Ter (pabrlos2(AT)yahoo.com), Nov 10 2005 %p A007723 L:=proc(p,n,k) options remember: local i: if (k=-1 or k=n) and n>=-1 and p>=1 then RETURN(1) elif p=0 and (k=-1 or k=n) then RETURN(1) elif p=0 and (k<>-1 and k<>n) then RETURN(0) elif p>=1 and n>=1 then RETURN(add(L(p-1,n,i)*L(p,i-1,k-1),i=k..n)) fi: end; M:=(p,n)->add(L(p,n,k),k=-1..n); seq(seq(M(n-i+1,i-1),i=0..n+1),n=-1..12); # first method (Pab Ter) %p A007723 L:=proc(p,n,k) options remember: local i: if (k=-1 or k=n) and n>=-1 and p>=1 then RETURN(1) elif p=0 and (k=-1 or k=n) then RETURN(1) elif p=0 and (k<>-1 and k<>n) then RETURN(0) elif p>=1 and n>=1 then RETURN(add(L(p-1,n,i)*L(p,i-1,k-1),i=k..n)) fi: end; M:=proc(p,n) options remember: local i: if n<1 and n>-2 and p>=0 then RETURN([1,2][n+2]) elif p=0 and n>=0 then RETURN(2) elif p>=1 and n>=1 then RETURN(1+add(L(p-1,n,i)*M(p,i-1),i=0..n)) fi: end; seq(seq(M(n-i+1,i-1),i=0..n+1),n=-1..12); # 2nd method (Pab Ter) %Y A007723 Cf. A003659, A011819, A011820, etc., A007065, A007625. %Y A007723 Adjacent sequences: A007720 A007721 A007722 this_sequence A007724 A007725 A007726 %Y A007723 Sequence in context: A053260 A014643 A118382 this_sequence A067437 A029315 A070080 %K A007723 nonn,nice,easy %O A007723 0,3 %A A007723 njas %E A007723 More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 10 2005 %I A067437 %S A067437 1,1,2,1,2,2,1,2,3,2,2,1,2,3,3,4,2,2,1,2,4,1,3,2,4,4,3,4,3,2,3,2,2,2,5, %T A067437 4,3,1,4,3,2,3,2,4,2,1,3,3,4,2,6,3,7,5,4,5,3,3,2,3,5,1,5,2,3,4,3,1,6,6, %U A067437 3,1,3,2,5,2,4,4,4,2,6,2,2,1,3,4,3,2,4,3 %N A067437 Number of distinct prime factors in binomial(2*n,n)+1. %Y A067437 Adjacent sequences: A067434 A067435 A067436 this_sequence A067438 A067439 A067440 %Y A067437 Sequence in context: A014643 A118382 A007723 this_sequence A029315 A070080 A131400 %K A067437 easy,nonn %O A067437 1,3 %A A067437 Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 23 2002 %I A029315 %S A029315 1,0,0,1,0,0,1,1,0,1,1,1,2,1,2,2,1,2,3,2,2,4,3,3,5,4,4, %T A029315 5,5,5,6,6,6,8,7,8,10,8,9,11,10,10,13,12,12,15,14,15,17, %U A029315 16,17,19,18,19,22,21,22,25,24,25,28,27,28,31,30,31,35 %N A029315 Expansion of 1/((1-x^3)(1-x^7)(1-x^11)(1-x^12)). %Y A029315 Adjacent sequences: A029312 A029313 A029314 this_sequence A029316 A029317 A029318 %Y A029315 Sequence in context: A118382 A007723 A067437 this_sequence A070080 A131400 A132749 %K A029315 nonn %O A029315 0,13 %A A029315 njas %I A070080 %S A070080 1,1,2,1,2,2,1,2,3,2,3,1,2,3,3,2,3,4,1,2,3,3,4,2,3,4,4,1,2,3,3,4,4,5,2, %T A070080 3,4,4,5,1,2,3,3,4,4,5,5,2,3,4,4,5,5,6,1,2,3,3,4,4,5,5,5,6,2,3,4,4,5,5, %U A070080 6,6,1,2,3,3,4,4,5,5,5,6,6,7,2,3,4,4,5,5 %N A070080 Smallest side of integer triangles [a(n)<=A070081(n)<=A070082(n)], sorted by perimeter, lexicographically ordered. %C A070080 a(n) + A070081(n) + A070082(n) = A070083(n). %H A070080 R. Zumkeller, Integer-sided triangles %Y A070080 Cf. A070084, A070085, A070086, A055594, A069597, A046128. %Y A070080 Adjacent sequences: A070077 A070078 A070079 this_sequence A070081 A070082 A070083 %Y A070080 Sequence in context: A007723 A067437 A029315 this_sequence A131400 A132749 A078498 %K A070080 nonn %O A070080 1,3 %A A070080 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002 %I A131400 %S A131400 1,2,1,2,2,1,2,3,3,1,2,3,6,3,1,2,4,7,7,4,1,2,4,11,8,11,4,12,5,12,15,15, %T A131400 12,5,1,2,5,17,16,30,16,17,5,1,2,6,18,27,36,36,27,18,6,1 %N A131400 A046854 + A065941 - I(Identity matrix). %C A131400 Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67,...). %e A131400 First few rows of the triangle are: %e A131400 1; %e A131400 2, 1; %e A131400 2, 2, 1; %e A131400 2, 3, 3, 1; %e A131400 2, 3, 6, 3, 1; %e A131400 2, 4, 7, 7, 4, 1; %e A131400 2, 4, 11, 8, 11, 4, 1; %e A131400 ... %Y A131400 Cf. A046854, A065941, A001595. %Y A131400 Adjacent sequences: A131397 A131398 A131399 this_sequence A131401 A131402 A131403 %Y A131400 Sequence in context: A067437 A029315 A070080 this_sequence A132749 A078498 A131240 %K A131400 nonn,tabl %O A131400 0,2 %A A131400 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 06 2007 %I A132749 %S A132749 1,2,1,2,2,1,2,3,3,1,2,4,6,4,1,2,5,10,10,5,1,2,6,15,20,15,6,1,2,7,21,35, %T A132749 35,21,7,1,2,8,28,56,70,56,28,8,1,2,9,36,84,126,126,84,36,9,1 %N A132749 A103451 * A007318. %C A132749 Row sums = A083318: (1, 3, 5, 9, 17, 33,...). %F A132749 A103451 * A007318 as infinite lower triangular matrices. Add (0, 1, 1, 1,...) to left column of A007318. %e A132749 First few rows of the triangle are: %e A132749 1; %e A132749 2, 1; %e A132749 2, 2, 1; %e A132749 2, 3, 3, 1; %e A132749 2, 4, 6, 4, 1; %e A132749 2, 5, 10, 10, 5, 1; %e A132749 ... %Y A132749 Cf. A103451, A083318, A007318. %Y A132749 Adjacent sequences: A132746 A132747 A132748 this_sequence A132750 A132751 A132752 %Y A132749 Sequence in context: A029315 A070080 A131400 this_sequence A078498 A131240 A107027 %K A132749 nonn,tabl %O A132749 0,2 %A A132749 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 28 2007 %I A078498 %S A078498 1,1,1,2,1,2,2,1,2,4,1,1,2,3,1,2,1,3,4,3,5,2,1,1,5,4,4,3,5,2,3,2,1,6,5, %T A078498 1,2,3,7,5,5,7,2,10,5,8,1,2,5,2,1,1,2,7,1,2,9,4,4,7,6,6,3,5,6,3,1,7,5,1, %U A078498 5,6,5,4,3,2,5,2,2,4,3,4,3,14,3,4,4,2,9,2,7,9,8,7,4,13 %N A078498 Let q(n) be the prime defined in A078497; sequence gives (q(n)-prime(n))/6. %F A078498 For n>4 a(n)=( min{p : p>prime(n), p and 2*prime(n)-p are primes} - prime(n) ) / 6. %e A078498 a(6)=1, a(25)=5. %Y A078498 Cf. A078496, A078497. %Y A078498 Adjacent sequences: A078495 A078496 A078497 this_sequence A078499 A078500 A078501 %Y A078498 Sequence in context: A070080 A131400 A132749 this_sequence A131240 A107027 A107030 %K A078498 nonn,easy %O A078498 5,4 %A A078498 Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), Nov 27 2002 %E A078498 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004 %I A131240 %S A131240 1,2,1,2,2,1,2,4,2,1,2,4,6,2,1,2,6,6,8,2,12,6,12,8,10,2,1,2,8,12,20,10, %T A131240 12,2,1,2,8,20,20,30,12,14,2,1,2,10,20,40,30,42,14,16,2,1 %N A131240 2*A046854 - I. %C A131240 Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67,...). A131241 = 3*A046854 - 2*I. %F A131240 2*A046854 - Identity matrix, where A046854 = Pascal's triangle with repeats by columns. %e A131240 First few rows of the triangle are: %e A131240 1; %e A131240 2, 1; %e A131240 2, 2, 1; %e A131240 2, 4, 2, 1; %e A131240 2, 4, 6, 2, 1; %e A131240 2, 6, 6, 8, 2, 1; %e A131240 2, 6, 12, 8, 10, 2, 1; %e A131240 ... %Y A131240 Cf. A001595, A046854, A131241. %Y A131240 Adjacent sequences: A131237 A131238 A131239 this_sequence A131241 A131242 A131243 %Y A131240 Sequence in context: A131400 A132749 A078498 this_sequence A107027 A107030 A050362 %K A131240 nonn,tabl %O A131240 0,2 %A A131240 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 21 2007 %I A107027 %S A107027 1,1,2,1,2,2,1,2,4,2,1,2,6,8,2,1,2,8,20,16,2,1,2,10,38,70,32,2,1,2,12, %T A107027 62,196,252,64,2,1,2,14,92,426,1062,924,128,2,1,2,16,128,792,3112,5948, %U A107027 3432,256,2,1,2,18,170,1326,7302,23686,34120,12870,512,2,1,2,20,218 %N A107027 Number triangle associated to the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0. %C A107027 As a number square read by anti-diagonals, the rows represent the row sums of the inverses of the Riordan arrays (1/(1+x),x/(1+x)^k), k>=0. The rows are then given by T(n,k)=(n-1)C(n*k,k)-(n-2)*sum{j=0..k, C(n*k,j)}. T(n,n) is A400000, T(n+1,n) is A000079, T(n+2,n) is A000984, T(n+3,n) is A047098. The reverse of this triangle is A107030. Row sums are A107028. Diagonal sums are A107029. %F A107027 Number triangle T(n, k)=if(k<=n, (n-k-1)C((n-k)*k, k)-(n-k-2)*sum{j=0..k, C((n-k)*k, j)}, 0). %e A107027 Triangle begins %e A107027 1; %e A107027 1,2; %e A107027 1,2,2; %e A107027 1,2,4,2; %e A107027 1,2,6,8,2; %e A107027 1,2,8,20,16,2; %Y A107027 Adjacent sequences: A107024 A107025 A107026 this_sequence A107028 A107029 A107030 %Y A107027 Sequence in context: A132749 A078498 A131240 this_sequence A107030 A050362 A095686 %K A107027 easy,nonn,tabl %O A107027 0,3 %A A107027 Paul Barry (pbarry(AT)wit.ie), May 09 2005 %I A107030 %S A107030 1,2,1,2,2,1,2,4,2,1,2,8,6,2,1,2,16,20,8,2,1,2,32,70,38,10,2,1,2,64,252, %T A107030 196,62,12,2,1,2,128,924,1062,426,92,14,2,1,2,256,3432,5948,3112,792, %U A107030 128,16,2,1,2,512,12870,34120,23686,7302,1326,170,18,2,1,2,1024,48620 %N A107030 Number triangle associated with the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0. %F A107030 Number triangle T(n, k)=(k-1)*C(k(n-k), n-k)-(k-2)*sum{j=0..n-k, C(k(n-k), j)} %e A107030 Triangle begins %e A107030 1; %e A107030 2,1; %e A107030 2,2,1; %e A107030 2,4,2,1; %e A107030 2,8,6,2,1; %e A107030 2,16,20,8,2,1; %Y A107030 Reversal of A107027. Row sums are A107028. Diagonal sums are A107031. Columns include A400000, A000079, A000984, A047098. %Y A107030 Adjacent sequences: A107027 A107028 A107029 this_sequence A107031 A107032 A107033 %Y A107030 Sequence in context: A078498 A131240 A107027 this_sequence A050362 A095686 A105258 %K A107030 easy,nonn,tabl %O A107030 0,2 %A A107030 Paul Barry (pbarry(AT)wit.ie), May 09 2005 %I A050362 %S A050362 1,1,1,1,2,1,2,2,1,3,1,2,1,4,2,3,2,5,2,1,4,1,4,2,6,3,2,5,1,4,3,8,4,2,6, %T A050362 2,6,1,4,10,4,5,1,4,3,8,2,8,2,5,12,4,6,1,2,6,4,10,3,10,2,6,15,6,8,1,2, %U A050362 8,4,5,12,1,4,4,12,3,4,8,9,18,8,10,2,3,10,4,6,15,2,6,5,16,1,4,4,10,12 %N A050362 Number of factorizations into distinct prime powers >1 indexed by prime signatures. %Y A050362 A050361(A025487). %Y A050362 Adjacent sequences: A050359 A050360 A050361 this_sequence A050363 A050364 A050365 %Y A050362 Sequence in context: A131240 A107027 A107030 this_sequence A095686 A105258 A109967 %K A050362 nonn %O A050362 1,5 %A A050362 Christian G. Bower (bowerc(AT)usa.net), Oct 15 1999. %E A050362 More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Nov 07 2001 %I A095686 %S A095686 1,1,1,2,1,2,2,1,3,1,2,2,1,3,1,3,2,2,1,4,2,2,3,1,4,1,3,2,2,2,1,2,2,4,1, %T A095686 4,1,3,3,2,1,5,3,2,3,1,4,2,4,2,2,1,6,1,2,3,2,4,1,3,2,4,1,6,1,2,3,3,2,4, %U A095686 1,5,2,1,6,2,2,2,4,1,6,2,3,2,2,2,6,1,3,3,1,4,1,4,4,2,1,6,1,4,2,5,1,4,2 %N A095686 Half the number of divisors of nonsquares (A000037). %C A095686 The first occurrence of n in the sequence corresponds to the nonsquare = A003680(n) = A005179(2n). %C A095686 Also number of unordered divisor pairs (d,n/d) for n=A000037. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 30 2004 %F A095686 a(n)=A000005(A000037(n))/2. %Y A095686 Adjacent sequences: A095683 A095684 A095685 this_sequence A095687 A095688 A095689 %Y A095686 Sequence in context: A107027 A107030 A050362 this_sequence A105258 A109967 A000119 %K A095686 nonn %O A095686 1,4 %A A095686 Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 05 2004 %E A095686 Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 09 2004 %I A105258 %S A105258 1,1,2,1,2,2,1,3,1,2,2,1,3,2,1,3,1,3,2,1,1,2,2,1,3,2,1,3,1,3,2,1,2,1,3, %T A105258 1,3,2,1,1,3,2,1,2,1,1,3,2,1,2,2,1,3,2,1,3,1,3,2,1,2,1,3,1,3,2,1,1,3,2, %U A105258 1,2,1,1,3,2,2,1,3,1,3,2,1,1,3,2,1,2,1,1,3,2,1,3,2,1,2,1,1,3,2,2,1,1,3 %N A105258 Second bi-Rauzy substitution that has the same characteristic digraph polynomial as the bi-Kenyon version: x^6-2x^4-2*x^3-x^2+2*x+1=0. %C A105258 Matrix is: M1 = {{0, 1, 0, 0, 0, 0}, {1, 0, 1, 0, 0, 0}, {1, 0,0, 0, 0, 0}, {1, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 1}, {0, 0, 0, 1, 0, 0}} Polynomial is: Det[M1 - x*IdentityMatrix[6]] NSolve[Det[M1 - x*IdentityMatrix[6]] == 0, x] %F A105258 1->{2}, 2->{1, 3}, 3->{1}, 4->{1, 5}, 5->{4, 6}, 6->{4} %e A105258 Triangle form is: %e A105258 {1} %e A105258 {1, 2} %e A105258 {1, 2, 2, 1, 3} %e A105258 {1, 2, 2, 1, 3, 2, 1, 3, 1, 3, 2, 1} %t A105258 s[1] = {2}; s[2] = {1, 3}; s[3] = {1}; s[4] = {1, 5}; s[5] = {4, 6}; s[6] = {4}; t[a_] := Join[a, Flatten[s /@ a]]; p[0] = {1}; p[1] = t[{1}]; p[n_] := t[p[n - 1]] aa = Flatten[Table[p[n], {n, 0, 6}]] %Y A105258 Cf. A073058, A105111. %Y A105258 Adjacent sequences: A105255 A105256 A105257 this_sequence A105259 A105260 A105261 %Y A105258 Sequence in context: A107030 A050362 A095686 this_sequence A109967 A000119 A097368 %K A105258 nonn,uned %O A105258 0,3 %A A105258 Roger Bagula (rlbagulatftn(AT)yahoo.com), Apr 14 2005 %I A109967 %S A109967 0,1,0,1,1,1,1,2,1,2,2,1,3,2,2,2,3,1,3,2,2,2,1,2,1,2,0,2,0,1,0,2,0,0,2, %T A109967 1,2,0,2,0,3,0,3,2,0,4,1,3,1,4,0,4,1,4,2,0,4,0,4,0,4,0,4,0,5,2,0,6,0,5, %U A109967 1,6,0,7,1,7,4,0,8,1,7,1,8,0,8,1,8,4,0,9,0,8,0,9,1,9,0,10,4,1,11,1 %V A109967 0,1,0,1,1,1,1,2,1,2,2,1,3,2,2,2,3,1,3,2,2,2,1,2,1,2,0,2,0,1,0,2,0,0,2,-1,2,0,2,0,3,0, %W A109967 3,2,0,4,1,3,1,4,0,4,1,4,2,0,4,0,4,0,4,0,4,0,5,2,0,6,0,5,1,6,0,7,1,7,4,0,8,1,7,1,8,0,8, %X A109967 1,8,4,0,9,0,8,0,9,-1,9,0,10,4,-1,11,-1 %N A109967 First differences of A088670. %C A109967 a(n) = A088670(n+1) - A088670(n). %Y A109967 Cf. A109969. %Y A109967 Adjacent sequences: A109964 A109965 A109966 this_sequence A109968 A109969 A109970 %Y A109967 Sequence in context: A050362 A095686 A105258 this_sequence A000119 A097368 A109699 %K A109967 sign %O A109967 1,8 %A A109967 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 06 2005 %I A000119 M0101 N0037 %S A000119 1,1,1,2,1,2,2,1,3,2,2,3,1,3,3,2,4,2,3,3,1,4,3,3,5,2,4,4,2,5,3,3,4,1,4, %T A000119 4,3,6,3,5,5,2,6,4,4,6,2,5,5,3,6,3,4,4,1,5,4,4,7,3,6,6,3,8,5,5,7,2,6,6, %U A000119 4,8,4,6,6,2,7,5,5,8,3,6,6,3,7,4,4,5,1,5,5,4,8,4,7,7,3,9,6,6,9,3,8,8,5 %N A000119 Number of representations of n as a sum of distinct Fibonacci numbers. %C A000119 Number of partitions into distinct Fibonacci parts (1 counted as single Fibonacci number) %C A000119 Inverse Euler transform of sequence has generating function sum_{n>1} x^F(n)-x^{2F(n)} where F() are the Fibonacci numbers. %C A000119 A065033(n) = a(A000045(n)). %D A000119 J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated to Aldo De Luca on the occasion of his 60-th anniversary. %D A000119 M. Bicknell-Johnson, pp. 53-60 in 'Applications of Fibonacci Numbers', volume 8, ed: F T Howard, Kluwer (1999); see Theorem 3. %D A000119 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 54. %D A000119 D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, Fib. Quart., 4 (1966), 289-306 and 322. %H A000119 T. D. Noe, Table of n, a(n) for n = 0..6765 %H A000119 Jean Berstel, Home Page %H A000119 Ron Knott Sumthing about Fibonacci Numbers %H A000119 J. Shallit, Number theory and formal languages, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller, and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570. (Eq. 9.2.) %F A000119 a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), b(k) = Sum_{f} (-1)^(k/f+1)*f, where the last sum is taken over all Fibonacci numbers f dividing k. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 28 2002 %F A000119 a(n)= 1, if n=0, 1, 2; a(n)= a(fib(i-2)+k)+a(k) if n>2 and 0<=k<=fib(i-3); a(n)= 2*a(k) if n>2 and fib(i-3)<=k<=fib(i-2); a(n)= a(fib(i+1)-2-k) otherwise where fib(i) is largest Fibonacci number (A000045) <= n and k=n-fib(i). [Bicknell-Johnson] - Ron Knott (ron(AT)ronknott.com), Dec 06 2004 %p A000119 with(combinat): p := product((1+x^fibonacci(i)), i=2..25): s := series(p,x,1000): for k from 0 to 250 do printf(`%d,`,coeff(s,x,k)) od: %t A000119 CoefficientList[ Normal@Series[ Product[ 1+z^Fibonacci[ k ], {k, 2, 13} ], {z, 0, 233} ], z ] %o A000119 (PARI) a(n)=local(A,m,f); if(n<0,0,A=1+x*O(x^n); m=2; while((f=fibonacci(m))<=n,A*=1+x^f; m++); polcoeff(A,n)) %Y A000119 Cf. A007000, A003107, A000121. Least inverse is A013583. %Y A000119 Adjacent sequences: A000116 A000117 A000118 this_sequence A000120 A000121 A000122 %Y A000119 Sequence in context: A095686 A105258 A109967 this_sequence A097368 A109699 A029283 %K A000119 nonn,nice %O A000119 0,4 %A A000119 njas %E A000119 More terms and Maple program from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000 %I A097368 %S A097368 1,1,2,1,2,2,1,3,2,2,3,1,3,3,2,4,2,3,4,1,4,3,3,5,2,4,4,2,5,3,4,5,1,5,4, %T A097368 3,6,3,5,5,2,6,4,4,2,6,5,3,7,4,5,6,1,7,5,4,7,3,6,6,3,8,5,5,7,2,7,6,4,8, %U A097368 4,6,7,2,8,6,5,8,3,7,7,4,9,5,6,8,1,8,7,5,9,4,7,8,3,9,6,6,9,3,8,8,5,10,5 %N A097368 Least term in row n of the Fibonacci regression array in A097367. %e A097368 Row 8 of the array in A097367 is 7 6 5 4 1 4 6, of which the least term is T(8,5)=1. %Y A097368 Cf. A000045, A097367, A097369. %Y A097368 Adjacent sequences: A097365 A097366 A097367 this_sequence A097369 A097370 A097371 %Y A097368 Sequence in context: A105258 A109967 A000119 this_sequence A109699 A029283 A116482 %K A097368 nonn,tabl %O A097368 2,3 %A A097368 Clark Kimberling (ck6(AT)evansville.edu), Aug 09 2004 %I A109699 %S A109699 1,0,0,1,0,0,1,0,1,1,0,1,1,1,1,1,2,1,2,2,1,3,2,2,4,2,4,4,3,5,4,5,6,5,7, %T A109699 6,8,8,7,11,9,10,13,10,14,14,14,17,16,19,19,20,24,21,27,27,27,33,30,35, %U A109699 38,36,44,42,47,51,50,58,57,63,68,66,79,76,82,92,88,101,104,107,120 %N A109699 Number of partitions of n into parts each equal to 3 mod 5. %F A109699 G.f.=1/product(1-x^(3+5j), j=0..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006 %e A109699 a(21)=3 since 21 = 18+3 = 13+8 = 3+3+3+3+3+3+3 %p A109699 g:=1/product(1-x^(3+5*j),j=0..25): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..80); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006 %Y A109699 Adjacent sequences: A109696 A109697 A109698 this_sequence A109700 A109701 A109702 %Y A109699 Sequence in context: A109967 A000119 A097368 this_sequence A029283 A116482 A029287 %K A109699 nonn %O A109699 0,17 %A A109699 Erich Friedman (efriedma(AT)stetson.edu), Aug 07 2005 %E A109699 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006 %I A029283 %S A029283 1,0,0,1,0,1,1,0,2,1,2,2,1,3,2,3,4,2,5,4,5,6,4,7,7,7,9, %T A029283 7,10,10,11,12,11,14,14,15,17,15,19,19,21,22,21,25,25,27, %U A029283 29,27,33,32,35,37,35,41,41,43,47,44,51,51,54,57,55,62 %N A029283 Expansion of 1/((1-x^3)(1-x^5)(1-x^8)(1-x^10)). %Y A029283 Adjacent sequences: A029280 A029281 A029282 this_sequence A029284 A029285 A029286 %Y A029283 Sequence in context: A000119 A097368 A109699 this_sequence A116482 A029287 A055184 %K A029283 nonn %O A029283 0,9 %A A029283 njas %I A116482 %S A116482 1,1,1,1,2,1,2,2,1,3,3,1,4,4,2,1,5,6,3,1,6,8,5,2,1,8,11,7,3,1,10,14,10, %T A116482 5,2,1,12,19,14,7,3,1,15,24,19,11,5,2,1,18,31,26,15,7,3,1,22,39,34,21, %U A116482 11,5,2,1,27,49,45,29,15,7,3,1,32,61,58,39,22,11,5,2,1,38,76,75,52,30 %N A116482 Triangle read by rows: T(n,k) is the number of partitions of n having k even parts (n>=0, 0<=k<=floor(n/2)). %C A116482 Row n has 1+floor(n/2) terms. Row sums are the partition numbers (A000041). Column 0 yields A000009. Column 1 yields A038348. Sum(k*T(n,k),k=0..floor(n/2))=A066898(n). %F A116482 G.f.= G(t,x)=1/product((1-x^(2j-1))(1-tx^(2j)),j=1..infinity). %e A116482 T(7,2)=3 because we have [4,2,1],[3,2,2], and [2,2,1,1,1]. %e A116482 Triangle starts: %e A116482 1; %e A116482 1; %e A116482 1,1; %e A116482 2,1; %e A116482 2,2,1; %e A116482 3,3,1; %e A116482 4,4,2,1; %e A116482 5,6,3,1; %p A116482 g:=1/product((1-x^(2*j-1))*(1-t*x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser,x^n) od: for n from 0 to 18 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form %Y A116482 Cf. A000041, A066898, A103919, A038348. %Y A116482 Adjacent sequences: A116479 A116480 A116481 this_sequence A116483 A116484 A116485 %Y A116482 Sequence in context: A097368 A109699 A029283 this_sequence A029287 A055184 A035388 %K A116482 nonn,tabf %O A116482 0,5 %A A116482 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006 %I A029287 %S A029287 1,0,0,1,0,1,1,0,1,2,1,2,2,1,3,3,2,3,4,3,5,5,4,6,6,6,7, %T A029287 8,7,9,10,9,11,12,11,13,15,13,16,17,16,19,20,19,22,24,22, %U A029287 26,27,26,30,31,30,34,36,35,39,40,40,44,46,45,49,52,51 %N A029287 Expansion of 1/((1-x^3)(1-x^5)(1-x^9)(1-x^11)). %Y A029287 Adjacent sequences: A029284 A029285 A029286 this_sequence A029288 A029289 A029290 %Y A029287 Sequence in context: A109699 A029283 A116482 this_sequence A055184 A035388 A077053 %K A029287 nonn %O A029287 0,10 %A A029287 njas %I A055184 %S A055184 1,1,1,1,1,1,1,2,1,2,2,1,3,3,3,3,1,2,2,2,2,3,4,2,3,4,2,4,2,5,2,1,3,6,2, %T A055184 4,3,4 %N A055184 Number of new numbers in n-th segment of A055177; see line %e of A055177. %C A055184 Also, the number of new numbers in n-th segment of A055193. %Y A055184 Adjacent sequences: A055181 A055182 A055183 this_sequence A055185 A055186 A055187 %Y A055184 Sequence in context: A029283 A116482 A029287 this_sequence A035388 A077053 A060438 %K A055184 nonn %O A055184 1,8 %A A055184 Clark Kimberling (ck6(AT)evansville.edu), Apr 27 2000 %I A035388 %S A035388 0,0,1,0,1,1,0,1,2,1,2,2,1,3,4,2,4,5,3,7,7,5,9,9,8,13,13,11,17,18,16, %T A035388 24,24,22,32,32,31,43,42,42,56,56,57,74,74,75,96,96,100,125,124,130, %U A035388 160,161,171,205,205,219,261,264,282,330,334,359,418,424,456,524,533 %N A035388 Number of partitions of n into parts 6k+3 or 6k+5. %Y A035388 Adjacent sequences: A035385 A035386 A035387 this_sequence A035389 A035390 A035391 %Y A035388 Sequence in context: A116482 A029287 A055184 this_sequence A077053 A060438 A106190 %K A035388 nonn %O A035388 1,9 %A A035388 Olivier Gerard (ogerard(AT)ext.jussieu.fr) %I A077053 %S A077053 1,1,1,1,2,1,2,2,1,4,1,1,1,1,2,1,1,2,1,4,1,1,3,1,4,1,1,1,1,1,1,8,1,4,5, %T A077053 1,8,4,1,1,1,1,1,8 %N A077053 Greatest common divisor of indecomposable Wallis pairs. %C A077053 Terms > 1 show that (x,y) need not be a Wallis pair if (cx,cy) is a Wallis pair. %F A077053 a(n) = gcd(A075768(n), A075769(n)) %Y A077053 Cf. A072182, A072186, A075768, A075769. %Y A077053 Adjacent sequences: A077050 A077051 A077052 this_sequence A077054 A077055 A077056 %Y A077053 Sequence in context: A029287 A055184 A035388 this_sequence A060438 A106190 A029290 %K A077053 nonn %O A077053 0,5 %A A077053 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 22 2002 %I A060438 %S A060438 1,2,1,2,2,1,4,2,2,1,7,2,2,2,1,12,4,2,2,2,1,16,7,2,2,2,2,1,32,12,4, %T A060438 2,2,2,2,1 %N A060438 Triangle T(n,k), 1 <= k <= n, giving maximal size of binary code of length n and covering radius k. %D A060438 G. D. Cohen et al., Covering Codes, North-Holland, 1997, p. 166. %D A060438 W. Haas, Binary and ternary codes of covering radius one ..., Discrete Math., 245 (2002), 161-178. %H A060438 Index entries for sequences related to covering codes %e A060438 1; 2,1; 2,2,1; 4,2,2,1; 7,2,2,2,1; ... %Y A060438 Columns give A000983, A029866. Cf. A060439, A060440. %Y A060438 Adjacent sequences: A060435 A060436 A060437 this_sequence A060439 A060440 A060441 %Y A060438 Sequence in context: A055184 A035388 A077053 this_sequence A106190 A029290 A115311 %K A060438 nonn,tabl,nice %O A060438 1,2 %A A060438 njas, Apr 07 2001 %E A060438 Next term, T(9,1), is in range 55-62. %I A106190 %S A106190 1,2,1,2,2,1,4,2,2,1,10,4,2,2,1,28,10,4,2,2,1,84,28,10,4,2,2,1,264,84, %T A106190 28,10,4,2,2,1,858,264,84,28,10,4,2,2,1,2860,858,264,84,28,10,4,2,2,1, %U A106190 9724,2860,858,264,84,28,10,4,2,2,1,33592,9724,2860,858,264,84,28,10,4 %V A106190 1,-2,1,-2,-2,1,-4,-2,-2,1,-10,-4,-2,-2,1,-28,-10,-4,-2,-2,1,-84,-28,-10,-4,-2,-2,1, %W A106190 -264,-84,-28,-10,-4,-2,-2,1,-858,-264,-84,-28,-10,-4,-2,-2,1,-2860,-858,-264,-84,-28, %X A106190 -10,-4,-2,-2,1,-9724,-2860,-858,-264,-84,-28,-10,-4,-2,-2,1,-33592,-9724,-2860,-858 %N A106190 Triangle read by rows: T(n,k)=binomial(2(n-k),n-k)/(1-2(n-k)). %C A106190 Sequence array for expansion of sqrt(1-4x). %C A106190 Row sums are A106191. Diagonal sums are A106192. Sequence array for A002420. Inverse of number triangle A106187. %C A106190 Riordan array (sqrt(1-4x),x). %e A106190 Triangle begins %e A106190 1; %e A106190 -2,1; %e A106190 -2,-2,1; %e A106190 -4,-2,-2,1; %e A106190 -10,-4,-2,-2,1; %e A106190 -28,-10,-4,-2,-2,1; %t A106190 T[n_, k_] := Binomial[2(n - k), n - k]/(1 - 2(n - k)); Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 25 2005) %Y A106190 Adjacent sequences: A106187 A106188 A106189 this_sequence A106191 A106192 A106193 %Y A106190 Sequence in context: A035388 A077053 A060438 this_sequence A029290 A115311 A035436 %K A106190 easy,sign,tabl %O A106190 0,2 %A A106190 Paul Barry (pbarry(AT)wit.ie), Apr 24 2005 %E A106190 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 25 2005 %I A029290 %S A029290 1,0,0,1,0,1,1,0,1,1,2,1,2,2,1,4,2,2,4,2,5,4,4,5,5,7,5, %T A029290 7,7,6,11,7,9,11,9,13,12,12,13,14,17,14,18,17,17,23,19, %U A029290 21,24,22,27,26,27,28,30,33,31,35,35,35,43,38,41,45,43 %N A029290 Expansion of 1/((1-x^3)(1-x^5)(1-x^10)(1-x^12)). %Y A029290 Adjacent sequences: A029287 A029288 A029289 this_sequence A029291 A029292 A029293 %Y A029290 Sequence in context: A077053 A060438 A106190 this_sequence A115311 A035436 A035369 %K A029290 nonn %O A029290 0,11 %A A029290 njas %I A115311 %S A115311 0,2,1,2,2,1,4,2,3,2,22,1,8,2,29,2,42,1,76,2,55,2,398,1,144,2,521,2,754, %T A115311 1,1364,2,987,2,7142,1,2584,2,9349,2,13530,1,24476,2,17711,2,128158,1, %U A115311 46368,2,167761,2,242786,1,439204,2,317811,2,2299702,1 %N A115311 GCD(Lucas(n)-1,Fibonacci(n)-1). %C A115311 Lucas(1)=1, Lucas(2)=3, Lucas(n>2)=Lucas(n-1)+Lucas(n-2) A000032. %e A115311 a(15)=29 since F(15)-1 =3*7*29 and L(15)-1=29*49 %t A115311 lucas[1]=1; lucas[2]=3; lucas[n_]:= lucas[n]= lucas[n-1] + lucas[n-2]; Table[GCD[lucas[i]-1, Fibonacci[i]-1], {i, 60}] %Y A115311 Cf. A000032, A000045, A111956, A115312, A115313, A115314. %Y A115311 Adjacent sequences: A115308 A115309 A115310 this_sequence A115312 A115313 A115314 %Y A115311 Sequence in context: A060438 A106190 A029290 this_sequence A035436 A035369 A129719 %K A115311 easy,nonn %O A115311 1,2 %A A115311 Giovanni Resta (g.resta(AT)iit.cnr.it), Jan 20 2006 %I A035436 %S A035436 0,0,1,0,1,1,0,1,1,2,1,2,2,1,4,2,3,4,3,6,4,6,6,7,9,7,11,9,12,15,12,17, %T A035436 16,20,21,23,26,25,33,32,36,40,41,49,49,57,58,66,72,75,87,86,100,106, %U A035436 113,126,130,148,151,172,180,191,216,219,248,258,280,304,318,355,364 %N A035436 Number of partitions of n into parts 7k+3 or 7k+5. %Y A035436 Adjacent sequences: A035433 A035434 A035435 this_sequence A035437 A035438 A035439 %Y A035436 Sequence in context: A106190 A029290 A115311 this_sequence A035369 A129719 A062602 %K A035436 nonn %O A035436 1,10 %A A035436 Olivier Gerard (ogerard(AT)ext.jussieu.fr) %I A035369 %S A035369 0,0,1,0,1,1,0,2,1,2,2,1,4,2,4,5,2,8,5,7,10,5,14,11,12,19,11,24,21,22, %T A035369 33,22,41,38,37,58,40,67,68,63,95,73,108,114,105,155,124,173,188,171, %U A035369 246,208,270,303,274,385,338,418,477,435,590,539,640,738,676,898,840 %N A035369 Number of partitions of n into parts 5k or 5k+3. %Y A035369 Adjacent sequences: A035366 A035367 A035368 this_sequence A035370 A035371 A035372 %Y A035369 Sequence in context: A029290 A115311 A035436 this_sequence A129719 A062602 A123148 %K A035369 nonn %O A035369 1,8 %A A035369 Olivier Gerard (ogerard(AT)ext.jussieu.fr) %I A129719 %S A129719 1,1,1,2,1,2,2,1,4,3,1,4,5,3,1,8,8,4,1,8,12,9,4,1,16,20,13,5,1,16,28,25, %T A129719 14,5,1,32,48,38,19,6,1,32,64,66,44,20,6,1,64,112,104,63,26,7,1,64,144, %U A129719 168,129,70,27,7,1,128,256,272,192,96,34,8,1,128,320,416,360,225,104,35 %N A129719 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 0's in odd positions (0<=k<=ceil(n/2)). A Fibonacci binary word is a binary word having no 00 subword. %C A129719 Row n has 1+ceil(n/2) terms. Row sums are the Fibonacci numbers (A000045). T(2n,k)=T(2n-1,k)+T(2n-2,k) (n>=1). T(2n,k)=A129721(2n,k). Sum(k*T(n,k), 0<=k<=ceil(n/2))=A129720(n). %F A129719 G.f.=G(t,z)=(1+z)(1+tz-tz^2)/[1-(2+t)z^2+tz^4]. The trivariate generating function H(t,s,z), where t marks number of 0's in odd position and s marks number of 0's in even position, is given by H(t,s,z)=[1+(1+t)z-tsz^3]/[1-(1+t+s)z^2+tsz^4]. %e A129719 T(6,2)=4 because we have 110101, 011101, 010110, and 010111. %e A129719 Triangle starts: %e A129719 1; %e A129719 1,1; %e A129719 2,1; %e A129719 2,2,1; %e A129719 4,3,1; %e A129719 4,5,3,1; %e A129719 8,8,4,1; %p A129719 G:=(1+z)*(1+t*z-t*z^2)/(1-(2+t)*z^2+t*z^4): Gser:=simplify(series(G,z=0,20)): for n from 0 to 17 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 17 do seq(coeff(P[n],t,j),j=0..ceil(n/2)) od; # yields sequence in triangular form %Y A129719 Cf. A000045, A129720, A129721. %Y A129719 Adjacent sequences: A129716 A129717 A129718 this_sequence A129720 A129721 A129722 %Y A129719 Sequence in context: A115311 A035436 A035369 this_sequence A062602 A123148 A134997 %K A129719 nonn,tabf %O A129719 0,4 %A A129719 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2007 %I A062602 %S A062602 0,0,1,1,0,2,1,2,2,1,4,3,3,3,4,2,6,3,5,4,6,3,8,3,7,4,9,5,9,4,8,7,9,4, %T A062602 11,3,11,9,10,6,12,5,11,8,12,7,14,5,13,7,15,9,15,6,14,10,16,9,16,5,15, %U A062602 13,16,8,18,6,18,15,17,9,19,8,18,12,19,11,21,7,21,14,20,13,22,7,21,14 %N A062602 Number of ways of writing n = p+c with p prime and c nonprime (1 or a composite number). %H A062602 Index entries for sequences related to Goldbach conjecture %e A062602 n = 22 has Floor[n/2] = 11 partitions of form n = a+b; 3 partitions are of prime+prime [3+19 = 5+17 = 11+11], 3 partitions are of prime+nonprime [2+20 = 7+15 = 13+9], 5 partitions are nonprime+nonprime [1+21 = 4+18 = 6+16 = 8+14 = 10+12]. So a(22) = 3. %Y A062602 Cf. A061358, A014092, A062610. %Y A062602 Adjacent sequences: A062599 A062600 A062601 this_sequence A062603 A062604 A062605 %Y A062602 Sequence in context: A035436 A035369 A129719 this_sequence A123148 A134997 A104605 %K A062602 nonn %O A062602 1,6 %A A062602 Labos E. (labos(AT)ana.sote.hu), Jul 04 2001 %I A123148 %S A123148 1,2,1,2,2,1,4,4,2,1,4,8,6,2,1,8,12,12,8,2,1,8,24,24,16,10,2,1,16,32,48, %T A123148 40,20,12,2,1,16,64,80,80,60,24,14,2,1,32,80,160,160,120,84,28,16,2,1, %U A123148 32,160,240,320,280,168,112,32,18,2,1 %V A123148 -1,-2,1,-2,2,-1,-4,4,-2,1,-4,8,-6,2,-1,-8,12,-12,8,-2,1,-8,24,-24,16,-10,2,-1,-16,32, %W A123148 -48,40,-20,12,-2,1,-16,64,-80,80,-60,24,-14,2,-1,-32,80,-160,160,-120,84,-28,16,-2,1, %X A123148 -32,160,-240,320,-280,168,-112,32,-18,2,-1 %N A123148 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p[n,x] defined by p[0,x]=-1, p[1,x]=x-2, p[n,x]=-xp[n-1,x]+2p[n-2,x] for n>=3 (0<=k<=n). %C A123148 Row sums yield -1,-1,-1,... . Alternating row sums yield the Jacobsthal sequence (A001045) with changed signs. %e A123148 -1 %e A123148 -2+x %e A123148 -2+2*x-x^2 %e A123148 -4+4*x-2*x^2+x^3 %e A123148 -4+8*x-6*x^2+2*x^3-x^4 %p A123148 p[0]:=-1: p[1]:=x-2: for n from 2 to 10 do p[n]:=sort(expand(-x*p[n-1]+2*p[n-2])) od: for n from 0 to 10 do seq(coeff(p[n],x,k),k=0..n) od; # yields sequence in triangular form %t A123148 a = -1; b = 2; p[0, x] = -1; p[1, x] = x - 2; p[k_, x_] := p[k, x] = a*x*p[k - 1, x] + b*p[k - 2, x] w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w] %Y A123148 Adjacent sequences: A123145 A123146 A123147 this_sequence A123149 A123150 A123151 %Y A123148 Sequence in context: A035369 A129719 A062602 this_sequence A134997 A104605 A138516 %K A123148 sign,tabl %O A123148 0,2 %A A123148 Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 01 2006 %E A123148 Edited by njas, Oct 29 2006 %I A134997 %S A134997 2,1,2,2,1,14,40,52,228,482 %N A134997 Number of dihedral primes with n digits. %H A134997 C. K. Caldwell, The Prime Glossary, Dihedral Prime %H A134997 C. Rivera, Source %Y A134997 See A048660 for another version. Cf. A038136, A048662. %Y A134997 Adjacent sequences: A134994 A134995 A134996 this_sequence A134998 A134999 A135000 %Y A134997 Sequence in context: A129719 A062602 A123148 this_sequence A104605 A138516 A026513 %K A134997 more,nonn %O A134997 1,1 %A A134997 Patrick Capelle (patrick.capelle@clearwire.be), Feb 06 2008 %I A104605 %S A104605 0,2,1,2,2,2,0,2,4,1,3,4,1,3,4,4,4,0,4,4,2,1,4,4,2,2,4,4,2,0,2,4,6,3,1, %T A104605 5,6,3,1,5,6,3,2,5,6,3,0,2,5,6,1,3,5,6,1,3,5,6,6,6,0,6,6,2,1,6,6,2,2,6, %U A104605 6,2,0,2,6,6,4,1,3,6,6,4,1,3,6,6,4,4,6,6,4,0,4,6,6,4,2,1,4,6,6,4 %V A104605 0,-2,1,-2,2,-2,0,2,-4,-1,3,-4,1,3,-4,4,-4,0,4,-4,-2,1,4,-4,-2,2,4,-4,-2,0,2,4,-6,-3, %W A104605 -1,5,-6,-3,1,5,-6,-3,2,5,-6,-3,0,2,5,-6,-1,3,5,-6,1,3,5,-6,6,-6,0,6,-6,-2,1,6,-6,-2,2, %X A104605 6,-6,-2,0,2,6,-6,-4,-1,3,6,-6,-4,1,3,6,-6,-4,4,6,-6,-4,0,4,6,-6,-4,-2,1,4,6,-6,-4 %N A104605 Triangle read by rows: row n gives list of powers of phi in the representation of the integer n as a sum of distinct nonconsecutive powers of the golden ratio. %H A104605 Eric Weisstein's World of Mathematics, Phi Number System %e A104605 0; -2, 1; -2, 2; -2, 0, 2; -4, -1, 3; -4, 1, 3; -4, 4; -4, 0, 4; ... %e A104605 phi^0, phi^(-2) + phi, phi^(-2) + phi^2, phi^0 + phi^(-2) + phi^2, ... %Y A104605 Cf. A055778, A105424. %Y A104605 Adjacent sequences: A104602 A104603 A104604 this_sequence A104606 A104607 A104608 %Y A104605 Sequence in context: A062602 A123148 A134997 this_sequence A138516 A026513 A106028 %K A104605 sign,nice,tabf %O A104605 1,2 %A A104605 Eric Weisstein (eric(AT)weisstein.com), Mar 17, 2005 %I A138516 %S A138516 1,2,1,2,2,2,1,0,4,2,5,2,0,8,2,8,3,2,14,6,14,6,4,24,12,24,11,4,40,16,38, %T A138516 16,5,62,24,60,24,10,94,40,91,38,18,144,62,136,57,24,214,88,201,82,30, %U A138516 308,122,288,117,48,440,180,410,168,74,624,262,578,238,96,874,356,804 %V A138516 1,2,1,2,2,-2,-1,0,-4,-2,5,2,0,8,2,-8,-3,-2,-14,-6,14,6,4,24,12,-24,-11,-4,-40,-16,38, %W A138516 16,5,62,24,-60,-24,-10,-94,-40,91,38,18,144,62,-136,-57,-24,-214,-88,201,82,30,308, %X A138516 122,-288,-117,-48,-440,-180,410,168,74,624,262,-578,-238,-96,-874,-356,804 %N A138516 McKay-Thompson series of class 10E for the Monster group with a(0) = 2. %F A138516 Expansion of q^(-1) * (psi(q) / psi(q^5))^2 in powers of q where psi() is a Ramanujan theta function. %F A138516 Expansion of ((eta(q^2) / eta(q^10))^2 * eta(q^5) / eta(q))^2 in powers of q. %F A138516 Euler transform of period 10 sequence [ 2, -2, 2, -2, 0, -2, 2, -2, 2, 0, ...]. %F A138516 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - v) * (v - 1) - 4 * v * (u - 1). %F A138516 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u - 5) * v * (v - 1) * (v - 5). %F A138516 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5 g(t) where q = exp(2 pi i t) and g() is g.f. for A138517. %F A138516 G.f.: (1/x) * (Product_{k>0} P(5,x^k) * P(10,x^k)^2)^(-2) where P(n,x) is the nth cyclotomic polynomial. %e A138516 1/q + 2 + q + 2*q^2 + 2*q^3 - 2*q^4 - q^5 - 4*q^7 - 2*q^8 + 5*q^9 + ... %o A138516 (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( (eta(x^2 + A) / eta(x^10 + A))^2 * eta(x^5 + A) / eta(x + A))^2, n))} %Y A138516 A058101(n) = a(n) unless n=0. Convolution inverse of A138519. Convolution square of A138532. %Y A138516 Adjacent sequences: A138513 A138514 A138515 this_sequence A138517 A138518 A138519 %Y A138516 Sequence in context: A123148 A134997 A104605 this_sequence A026513 A106028 A105307 %K A138516 sign %O A138516 -1,2 %A A138516 Michael Somos, Mar 23 2008 %I A026513 %S A026513 2,1,2,2,2,1,1,1,2,1,2,2,2,2,2,2,2,1,2,2,2,1,1,1,2,1,1,1,1,1, %T A026513 1,1,2,1,2,2,2,1,1,1,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2, %U A026513 2,2,2,2,2,1,2,2,2,1,1,1,2,1,2,2,2,2,2,2,2,1,2,2,2,1,1,1,2,1 %N A026513 t(3+3n), where t = A001285 (Thue-Morse sequence). %Y A026513 Adjacent sequences: A026510 A026511 A026512 this_sequence A026514 A026515 A026516 %Y A026513 Sequence in context: A134997 A104605 A138516 this_sequence A106028 A105307 A082070 %K A026513 nonn %O A026513 0,1 %A A026513 Clark Kimberling (ck6(AT)evansville.edu) %I A106028 %S A106028 0,0,2,1,2,2,2,1,1,1,2,2,4,4,4,1,1,2,2,1,4,4,4,3,3,3,3,3,4,4,4,2,2,3,3, %T A106028 3,3,3,4,3,3,3,3,3,3,3,5,2,2,2,2,2,2,2,7,5,5,5,5,5,5,5,7,2,2,2,2,3,3,3, %U A106028 4,2,3,2,2,4,4,4,5,2,3,2,2,2,2,2,3,3,6,6,6,5,5,6,6,4,5,4,4,4,4,5,5,4,4 %N A106028 Minimal number of editing steps (delete, insert or substitute) to transform the binary representation of n into that of A003714(n), the n-th fibbinary number. %C A106028 A014417(n) = A007088(A003714(n)). %H A106028 Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - njas] %H A106028 Eric Weisstein's World of Mathematics, Zeckendorf Representation %H A106028 Index entries for sequences related to binary expansion of n %F A106028 a(n) = LevenshteinDistance(A014417(n), A007088(n)). %Y A106028 Cf. A035517, A000045, A072649, A070939. %Y A106028 Adjacent sequences: A106025 A106026 A106027 this_sequence A106029 A106030 A106031 %Y A106028 Sequence in context: A104605 A138516 A026513 this_sequence A105307 A082070 A082071 %K A106028 nonn %O A106028 1,3 %A A106028 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2005 %I A105307 %S A105307 0,0,1,1,1,2,1,2,2,2,1,1,1,2,3,3,1,4,2,4,3,2,1,1,1,2,4,4,1,6,2,4,3,2,3,1,3,3,3,6,2, %T A105307 6,1,5,5,3,1,1,3,1,3,4,2,7,4,1,5,3,2,1,2,3,5,6,3,6,3,5,5,7,2,1,2,4,1,5,4,7,2,9,7, %U A105307 3,1,1,4,3,4,9,2,11,1,6,4,2,6,1,4,5,6,1,2,7,3,7,7,3,2,1,2,1,5,1,2,9,4,6,6,5,3 %V A105307 0,0,1,1,1,2,1,2,2,2,1,-1,1,2,3,3,1,4,2,4,3,2,1,-1,-1,2,4,4,1,6,2,4,3,2,3,-1,3,3,3,6,2, %W A105307 6,1,5,5,3,1,-1,3,-1,3,4,2,7,4,-1,5,3,2,-1,2,3,5,6,3,6,3,5,5,7,2,-1,2,4,-1,5,4,7,2,9,7, %X A105307 3,1,-1,4,3,4,9,2,11,-1,6,4,2,6,-1,4,5,6,-1,2,7,3,7,7,3,2,-1,2,-1,5,-1,2,9,4,6,6,5,3 %N A105307 Log base 2 of the number of divisors of Fibonacci[n] if that number is a power of 2, else -1. %C A105307 It appears that the number of divisors of most Fibonacci numbers is a power of 2. %e A105307 F(6)=8 has 4 divisors {1,2,4,8}, so a(6)=log base 2 of 4=2 %Y A105307 Cf. A081979. %Y A105307 Adjacent sequences: A105304 A105305 A105306 this_sequence A105308 A105309 A105310 %Y A105307 Sequence in context: A138516 A026513 A106028 this_sequence A082070 A082071 A082902 %K A105307 sign %O A105307 1,6 %A A105307 John W. Layman (layman(AT)math.vt.edu), May 03 2005 %I A082070 %S A082070 1,1,2,1,2,2,2,1,1,2,2,2,2,2,2,1,2,3,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2, %T A082070 1,2,2,2,2,2,2,2,2,2,2,2,2,3,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2, %U A082070 2,3,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,1,2,2,2,2,2 %N A082070 Least common prime-divisor of Phi[n] and Sigma[1,n]=A000203(n); a(n)=1 if no common prime-divisor was found. %t A082070 ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] f1[x_] := EulerPhi[x]; f2[x_] := DivisorSigma[1, x] Table[Min[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}] %Y A082070 Cf. A000010, A000203, A082061-A082072. %Y A082070 Adjacent sequences: A082067 A082068 A082069 this_sequence A082071 A082072 A082073 %Y A082070 Sequence in context: A026513 A106028 A105307 this_sequence A082071 A082902 A123926 %K A082070 nonn %O A082070 1,3 %A A082070 Labos E. (labos(AT)ana.sote.hu), Apr 07 2003 %I A082071 %S A082071 1,1,2,1,2,2,2,1,1,2,2,2,2,2,2,1,2,3,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2, %T A082071 1,2,2,2,2,2,2,2,2,2,2,2,2,3,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2, %U A082071 2,3,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,1,2,2,2,2,2 %N A082071 Least common prime-divisor of Phi[n]=A000010(n) and Sigma[2,n]=A001157(n); a(n)=1 if no common prime-divisor was found. %t A082071 ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] f1[x_] := EulerPhi[x]; f2[x_] := DivisorSigma[2, x] Table[Min[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}] %Y A082071 Cf. A000010, A001157, A082061-A082072. %Y A082071 Adjacent sequences: A082068 A082069 A082070 this_sequence A082072 A082073 A082074 %Y A082071 Sequence in context: A106028 A105307 A082070 this_sequence A082902 A123926 A082064 %K A082071 nonn %O A082071 1,3 %A A082071 Labos E. (labos(AT)ana.sote.hu), Apr 07 2003 %I A082902 %S A082902 1,1,2,1,2,2,2,1,1,2,2,2,2,2,4,1,2,1,2,2,4,2,2,2,1,2,4,2,2,4,2,1,4,2,4, %T A082902 1,2,2,4,2,2,4,2,2,2,2,2,2,1,1,4,2,2,4,4,2,4,2,2,4,2,2,2,1,4,4,2,2,4,4, %U A082902 2,1,2,2,2,2,4,4,2,2,1,2,2,4,4,2,4,2,2,2,4,2,4,2,4,2,2,1,2,1,2,4,2,2,8 %N A082902 a(n)=gcd[2^n, sigma(2,n)]=gcd[A000079(n), A001157(n)]. %Y A082902 Adjacent sequences: A082899 A082900 A082901 this_sequence A082903 A082904 A082905 %Y A082902 Sequence in context: A105307 A082070 A082071 this_sequence A123926 A082064 A082065 %K A082902 nonn,mult %O A082902 1,3 %A A082902 Labos E. (labos(AT)ana.sote.hu), Apr 22 2003 %I A123926 %S A123926 1,1,2,1,2,2,2,1,1,2,2,2,2,2,4,1,2,1,2,6,4,2,2,2,1,2,4,2,2,4,2,3,4,2,4, %T A123926 1,2,2,4,2,2,4,2,6,2,2,2,2,3,3,4,2,2,4,4,2,4,2,2,12,2,2,2,1,4,4,2,6,4,4, %U A123926 2,1,2,2,2,2,4,4,2,2,1,2,2,4,4,2,4,2,2,2,4,6,4,2,4,6,2,3,2,1,2,4,2,2,8 %N A123926 Greatest common divisor of sigma_k(n) for all k >= 1. %C A123926 Has the property that if gcd(n,m) = 1, then a(n)*a(m) divides a(n*m). First inequality is a(4) = 1, a(5) = 2, but a(20) = 6. It appears that a(n) also always divides sigma_0(n) = tau(n). %e A123926 For n=4, sigma_1(n) = 7, sigma_2(n) = 21, both divisible by 7, but sigma_3(n) = 73, which is not, so a(4) = 1. %Y A123926 Cf. A109974, A000005, A000203, A001157. %Y A123926 Adjacent sequences: A123923 A123924 A123925 this_sequence A123927 A123928 A123929 %Y A123926 Sequence in context: A082070 A082071 A082902 this_sequence A082064 A082065 A082055 %K A123926 nice,nonn %O A123926 1,3 %A A123926 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 21 2006 %I A082064 %S A082064 1,1,2,1,2,2,2,1,1,2,2,2,2,3,2,1,2,3,2,2,2,2,2,2,1,3,2,2,2,2,2,1,2,2,3, %T A082064 1,2,3,2,2,2,3,2,2,3,2,2,2,3,1,2,2,2,3,2,3,2,2,2,2,2,3,2,1,3,2,2,2,2,3, %U A082064 2,3,2,3,2,2,3,3,2,2,1,2,2,2,2,3,2,5,2,3,2,2,2,2,3,2,2,3,3,1,2,2,2,3,3 %N A082064 Greatest common prime-divisor of Phi[n] and Sigma[1,n]=A000203(n); a(n)=1 if no common prime-divisor was found. %t A082064 ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] f1[x_] := EulerPhi[n]; f2[x_] := DivisorSigma[1, x] Table[Max[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}] %Y A082064 Cf. A006530, A000203, A000010, A082061-A082066. %Y A082064 Adjacent sequences: A082061 A082062 A082063 this_sequence A082065 A082066 A082067 %Y A082064 Sequence in context: A082071 A082902 A123926 this_sequence A082065 A082055 A073812 %K A082064 nonn %O A082064 1,3 %A A082064 Labos E. (labos(AT)ana.sote.hu), Apr 07 2003 %I A082065 %S A082065 1,1,2,1,2,2,2,1,1,2,2,2,2,3,2,1,2,3,2,2,2,2,2,2,1,3,2,2,2,2,2,1,2,2,3, %T A082065 1,2,3,2,2,2,3,2,2,3,2,2,2,3,1,2,2,2,3,2,3,2,2,2,2,2,3,2,1,3,2,2,2,2,3, %U A082065 2,3,2,3,2,2,3,3,2,2,1,2,2,2,2,3,2,5,2,3,2,2,2,2,3,2,2,3,3,1,2,2,2,3,3 %N A082065 Greatest common prime-divisor of Phi[n]=A000010(n) and Sigma[2,n]=A001157(n); a(n)=1 if no common prime-divisor was found. %t A082065 ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] f1[x_] := EulerPhi[n]; f2[x_] := DivisorSigma[2, x] Table[Max[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}] %Y A082065 Cf. A006530, A001157, A000010, A082061-A082066. %Y A082065 Adjacent sequences: A082062 A082063 A082064 this_sequence A082066 A082067 A082068 %Y A082065 Sequence in context: A082902 A123926 A082064 this_sequence A082055 A073812 A009223 %K A082065 nonn %O A082065 1,3 %A A082065 Labos E. (labos(AT)ana.sote.hu), Apr 07 2003 %I A082055 %S A082055 1,1,2,1,2,2,2,1,1,2,2,2,2,6,2,1,2,3,2,2,2,2,2,2,1,6,2,2,2,2,2,1,2,2,6, %T A082055 1,2,6,2,2,2,6,2,2,6,2,2,2,3,1,2,2,2,6,2,6,2,2,2,2,2,6,2,1,6,2,2,2,2,6, %U A082055 2,3,2,6,2,2,6,6,2,2,1,2,2,2,2,6,2,10,2,6,2,2,2,2,6,2,2,3,6,1,2,2,2,6,6 %N A082055 Product of common prime-divisors[without multiplicity] of sigma[n] and phi[n]. %t A082055 ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Table[Apply[Times, Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 256}] %Y A082055 Cf. A000203, A000010, A082054. %Y A082055 Adjacent sequences: A082052 A082053 A082054 this_sequence A082056 A082057 A082058 %Y A082055 Sequence in context: A123926 A082064 A082065 this_sequence A073812 A009223 A110244 %K A082055 nonn %O A082055 1,3 %A A082055 Labos E. (labos(AT)ana.sote.hu), Apr 03 2003 %I A073812 %S A073812 1,1,2,1,2,2,2,1,1,2,2,3,2,4,4,1,2,2,2,2,3,2,2,3,1,4,2,3,2,4,2,1,3,2,8, %T A073812 1,2,4,4,2,2,6,2,3,4,2,2,3,2,1,4,2,2,4,4,8,3,2,2,4,2,4,3,1,6,3,2,2,3,8, %U A073812 2,2,2,4,3,3,6,8,2,2,1,2,2,4,3,4,4,6,2,4,4,3,3,2,8,3,2,2,6,1,2,4,2,4 %N A073812 Number of common divisors of sigma[n] and Phi[n]. %F A073812 a(n)=Card[Intersection[D[A000203(n)], D[A000010(n)]]]. %e A073812 n=36: Sigma[36]=91; Phi[36]=12; Intersection[{1,7,13,91},{1,2,3,4,6,12}]={1}, so a(36)=1. %t A073812 g1[x_] := Divisors[DivisorSigma[1, x]] g2[x_] := Divisors[EulerPhi[x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}] %Y A073812 Cf. A000010, A000203, A073802, A073808-A073811. %Y A073812 Adjacent sequences: A073809 A073810 A073811 this_sequence A073813 A073814 A073815 %Y A073812 Sequence in context: A082064 A082065 A082055 this_sequence A009223 A110244 A055188 %K A073812 nonn %O A073812 1,3 %A A073812 Labos E. (labos(AT)ana.sote.hu), Aug 13 2002 %I A009223 %S A009223 1,1,2,1,2,2,2,1,1,2,2,4,2,6,8,1,2,3,2,2,4,2,2,4,1,6,2,4,2,8,2,1,4,2,24, %T A009223 1,2,6,8,2,2,12,2,4,6,2,2,4,3,1,8,2,2,6,8,24,4,2,2,8,2,6,4,1,12,4,2,2,4, %U A009223 24,2,3,2,6,4,4,12,24,2,2,1,2,2,8,4,6,8,20,2,6,8,4,4,2,24,4,2,3,12,1,2,8 %N A009223 GCD(sigma(n), phi(n)). %Y A009223 Adjacent sequences: A009220 A009221 A009222 this_sequence A009224 A009225 A009226 %Y A009223 Sequence in context: A082065 A082055 A073812 this_sequence A110244 A055188 A084989 %K A009223 nonn %O A009223 1,3 %A A009223 David W. Wilson (davidwwilson(AT)comcast.net) %I A110244 %S A110244 1,1,0,2,1,2,2,2,1,1,2,3,1,6,0,4,1,1,5,2,1,3,2,1,1,11,5,4,1,1,8,4,1,3,2, %T A110244 5,1,1,1,5,1,14,8,3,1,5,5,3,1,1,1,8,1,3,8,4,1,3,5,3,1,28,5,2,1,5,14,3,1, %U A110244 3,8,5,1,3,1,8,1,1,11 %V A110244 1,1,0,-2,1,2,2,-2,1,-1,2,-3,1,6,0,-4,1,-1,5,-2,1,-3,2,-1,1,11,5,-4,1,-1,8,-4,1,-3,2, %W A110244 -5,1,-1,1,-5,1,14,8,-3,1,-5,5,-3,1,-1,-1,-8,1,-3,8,-4,1,-3,5,-3,1,28,5,-2,1,-5,14,-3, %X A110244 1,-3,8,-5,1,-3,-1,-8,1,1,11 %N A110244 Diagonal sums of a Jacobi number triangle. %C A110244 Diagonal sums of number triangle A110242. %F A110244 a(n)=sum{k=0..floor(n/2), Jacobi(n-k, 2n-4k+1)} %Y A110244 Adjacent sequences: A110241 A110242 A110243 this_sequence A110245 A110246 A110247 %Y A110244 Sequence in context: A082055 A073812 A009223 this_sequence A055188 A084989 A128538 %K A110244 easy,sign %O A110244 0,4 %A A110244 Paul Barry (pbarry(AT)wit.ie), Jul 17 2005 %I A055188 %S A055188 2,1,2,2,2,1,1,4,2,3,1,5,2,4,1,1,4,1,3,6,2,7,1,3,4,2,3,1,5,8,2,9,1,4,4, %T A055188 4,3,2,5,1,6,1,7,10,2,12,1,7,4,5,3,3,5,2,6,2,7,1,8,1,9,13,2,15,1,8,4,7, %U A055188 3,5,5,3,6,4,7,2,8,2,9,1,10,1,12,16,2,18 %N A055188 Cumulative counting sequence: method A (adjective,noun)-pairs with first term 2. %C A055188 Segments (as in %e): 2; 1,2; 2,2,1,1; 4,2,3,1; 5,2,4,1,1,4,1,3; ... %C A055188 Conjecture: every nonnegative integer occurs. %e A055188 Write 2, thus having 1 2, thus having 2 2's and 1 1, thus having 4 2's and 3 1's, etc. %Y A055188 Adjacent sequences: A055185 A055186 A055187 this_sequence A055189 A055190 A055191 %Y A055188 Sequence in context: A073812 A009223 A110244 this_sequence A084989 A128538 A115588 %K A055188 nonn %O A055188 1,1 %A A055188 Clark Kimberling (ck6(AT)evansville.edu), Apr 27 2000 %I A084989 %S A084989 2,1,2,2,2,1,2,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,2,2,1,2,1,2,1,4,2,2,1, %T A084989 2,2,1,1,2,1,2 %N A084989 Number of 4-year terms served by Presidents of the U.S.A. %C A084989 The counts include partial presidential terms of office for a(9), a(10), a(12), a(13), a(16), a(17), a(20), a(21), a(25), a(26), a(29), a(30), a(32), a(33), a(35), a(36), a(37) and a(38). Grover Cleveland was both the 22nd and 24th President, so a(22) and a(24) refer to his two non-consecutive terms. %D A084989 "U.S. Presidents, Vice Presidents, Congresses", The World Almanac and Book of Facts (2000 edition). New Jersey: Primedia Reference Inc., 1999. pp. 503-504. %e A084989 George Washington, the first American President, served two terms, hence a(1)=2. %Y A084989 Cf. A008745 (Birth years of Presidents), A072250 (Vice Presidents under each President). %Y A084989 Adjacent sequences: A084986 A084987 A084988 this_sequence A084990 A084991 A084992 %Y A084989 Sequence in context: A009223 A110244 A055188 this_sequence A128538 A115588 A105220 %K A084989 nonn %O A084989 1,1 %A A084989 Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 16 2003 %I A128538 %S A128538 0,1,1,2,1,2,2,2,1,2,1,1,2,2,3,1,2,1,3,1,2,2,3,3,2 %N A128538 Number of prime factors (with multiplicity) of lucky numbers. %C A128538 a(n) = 0 iff n = 1. a(n) = 1 iff n-th lucky number is prime iff A000959(n) is in A031157 Numbers that are both lucky and prime. a(n) > 1 iff n-th lucky number is composite iff A000959(n) is in A031879 Composite lucky numbers [technically, A031879 should not begin with 1]. a(n) = 2 iff n-th lucky number is semiprime iff A000959(n) is in A001358. a(n) = 3 iff n-th lucky number has 3 prime factors (with multiplicity) iff A000959(n) is in A014612. %F A128538 a(n) = A001222(A000959(n)). %Y A128538 Cf. A000040, A000959, A001222, A001358, A014612, A031157, A031879. %Y A128538 Adjacent sequences: A128535 A128536 A128537 this_sequence A128539 A128540 A128541 %Y A128538 Sequence in context: A110244 A055188 A084989 this_sequence A115588 A105220 A083654 %K A128538 easy,nonn %O A128538 1,4 %A A128538 Jonathan Vos Post (jvospost2(AT)yahoo.com), May 07 2007 %I A115588 %S A115588 1,1,1,1,2,1,2,2,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,2,2,1,2,1,3,1,2,2,2,2,2, %T A115588 1,2,2,3,1,3,1,2,3,2,1,2,2,2,2,2,1,2,2,3,2,2,1,3,1,2,3,2,2,3,1,2,2,3,1, %U A115588 2,1,2,3,2,2,3,1,2,2,2,1,3,2,2,2,3,1,3,2,2,2,2,2,3,1,2,3,2,1,3,1,3,3,2 %N A115588 Number of unique prime numbers necessary to represent a natural number n > 1. %C A115588 The sequence gives the number of unique prime numbers needed to represent a given natural number greater than or equal to 2. In order to do this, we must factor any subsequent composite number that may appear on the exponents of the next factorizations (i.e. 4 in 48=2^4*3), until only prime numbers are used. - Lucas Vieira Barbosa (dnukem(AT)gmail.com), Mar 15 2006 %C A115588 In this sequence, a(n)=1 if n is prime, or a power tower (tetration or iterated exponentiation) of a prime base (i.e. 2^2, 3^3^3^3, 7^7, etc). The sequence reaches a new boundary whenever n is a primorial number (factorial of primes). - Lucas Vieira Barbosa (dnukem(AT)gmail.com), Mar 15 2006 %e A115588 a(4)=1, since 4=2^2 and the only prime used was 2; a(30)=3 because 30=2*3*5, and three primes were necessary; a(65536)=1, since 65536=2^16=2^(2^4)=2^(2^(2^2)) and, again, only one prime was needed; a(1) would be undefined, so it is not included. %Y A115588 Cf. A000040, A001221, A002110. %Y A115588 Adjacent sequences: A115585 A115586 A115587 this_sequence A115589 A115590 A115591 %Y A115588 Sequence in context: A055188 A084989 A128538 this_sequence A105220 A083654 A029428 %K A115588 nonn %O A115588 2,5 %A A115588 Lucas Vieira Barbosa (dnukem(AT)gmail.com), Mar 09 2006 %I A105220 %S A105220 1,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,2, %T A105220 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2, %U A105220 2,2,1,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 %N A105220 Trajectory of 1 under the morphism 1->{1,2,1}, 2->{2,2,2}. %C A105220 Dekking substitution for the Cantor set: characteristic polynomial=x^3+5*x^2-6*x. %C A105220 This substitution is useful for computing the devil's staircase by bb=aa/. 1->1/3/. 2->0 /. 3->0; ListPlot[FoldList[Plus, 0, bb], PlotRange -> All, PlotJoined -> True, Axes ->False]; %D A105220 F. M. Dekking, "Recurrent Sets",Advances in Mathematics, vol. 44, no. 1, 1982, page 99, section 4.15 %F A105220 1->{1, 2, 1}, 2->{2, 2, 2}, 3->{} (* Null substitution added in order to get characteristic polynomial*) %t A105220 Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {2, 2, 2}, 3 -> {}} &], {1}, 5]] %Y A105220 Adjacent sequences: A105217 A105218 A105219 this_sequence A105221 A105222 A105223 %Y A105220 Sequence in context: A084989 A128538 A115588 this_sequence A083654 A029428 A101422 %K A105220 nonn %O A105220 0,2 %A A105220 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 29 2005 %I A083654 %S A083654 1,1,1,2,1,2,2,2,1,2,1,2,2,3,2,2,1,2,2,2,2,2,2,3,2,3,2,3,2,3,2,2,1,2,2, %T A083654 2,1,2,2,2,2,2,2,2,2,2,3,3,2,3,3,2,2,3,2,4,2,3,2,3,2,3,2,2,1,2,2,2,2,2, %U A083654 2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,3,3,3,3,2,3,3,2,2,3,2,2,2 %N A083654 Consider the binary Champernowne sequence (A030190): number of successive numbers to be concatenated beginning with A083653(n) such that in binary representation n is contained. %C A083654 a(2^k)=1, see A083655 for all numbers m with a(m)=1; %e A083654 n=24: '11000'=24 is a suffix of the concatenation of the first 8 numbers: '0'1'10'11'100'101'110'111'1000', therefore a(24)=2 and A083653(24)=7. %Y A083654 Cf. A030304, A007088. %Y A083654 Adjacent sequences: A083651 A083652 A083653 this_sequence A083655 A083656 A083657 %Y A083654 Sequence in context: A128538 A115588 A105220 this_sequence A029428 A101422 A070304 %K A083654 nonn %O A083654 0,4 %A A083654 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 01 2003 %I A029428 %S A029428 1,0,0,0,0,0,1,0,0,1,1,1,1,0,0,1,1,1,2,1,2,2,2,1,2,1,2, %T A029428 3,3,3,4,3,3,4,3,3,5,4,5,6,6,5,7,5,6,7,7,7,9,8,9,10,9,9, %U A029428 11,10,11,12,12,12,15,13,14,15,15,15,18,16,17,19,19,19 %N A029428 Expansion of 1/((1-x^6)(1-x^9)(1-x^10)(1-x^11)). %Y A029428 Adjacent sequences: A029425 A029426 A029427 this_sequence A029429 A029430 A029431 %Y A029428 Sequence in context: A115588 A105220 A083654 this_sequence A101422 A070304 A083952 %K A029428 nonn %O A029428 0,19 %A A029428 njas %I A101422 %S A101422 1,1,1,2,1,2,2,2,1,2,1,3,2,3,1,2,3,2,2,3,1,2,3,3,2,3,1,2,3,3,2,3,3,2,3, %T A101422 3,2,3,3,2,1,3,2,3,1,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2, %U A101422 3,3,2,3,3,2,3,3,2,3,1,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,2,3,2,3,3,2,2,3 %N A101422 Minimal number of primes needed to sum to Fibonacci(n). %F A101422 a(n) = A051034(A000045(n)). %e A101422 a(5)=1 because Fibonacci(5)=5 is a prime. %e A101422 a(6)=2 because Fibonacci(6)=8 = 3+5. %e A101422 a(7)=1 because Fibonacci(7)=13 is a prime. %e A101422 a(14)=3 because Fibonacci(14)=377 = 2+2+373. %Y A101422 Cf. A000045, A051034, A001605, A102106, A102107. %Y A101422 Adjacent sequences: A101419 A101420 A101421 this_sequence A101423 A101424 A101425 %Y A101422 Sequence in context: A105220 A083654 A029428 this_sequence A070304 A083952 A043529 %K A101422 nonn %O A101422 3,4 %A A101422 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jan 17 2005 %E A101422 Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jan 18 2005 %I A070304 %S A070304 1,2,1,2,2,2,1,2,1,6,1,2,1,2,3,2,2,2,1,7,1,2,1,2,2,4,1,2,1,8,1,2,1,3,2, %T A070304 2,2,2,2,7,1,2,1,2,3,2,1,2,1,7,2,4,1,2,2,2,1,4,1,9,1,2,1,2,5,2,1,4,1,7, %U A070304 1,2,1,3,3,2,1,5,1,7,1,3,1,2,4,2,1,2,1,8,1,2,1,2,2,2,1,2,1,8,2,4,1,4,3 %N A070304 a(n) = number of times n^2*k^2/(n^2+k^2) is an integer as k ranges over 0, 1, 2, ... %C A070304 A018892(n) gives the number of integers of the form nk/(n+k). %o A070304 (PARI) for(n=1,150,print1(sum(i=0,n^2,if((n^2*i^2)%(n^2+i^2),0,1)),",")) %Y A070304 Cf. A066451. %Y A070304 Adjacent sequences: A070301 A070302 A070303 this_sequence A070305 A070306 A070307 %Y A070304 Sequence in context: A083654 A029428 A101422 this_sequence A083952 A043529 A080942 %K A070304 easy,nonn %O A070304 1,2 %A A070304 Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2002 %I A083952 %S A083952 1,2,1,2,2,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,1,2,1, %T A083952 2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,1,2,2,2,1,2,1,2,2,2, %U A083952 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 %N A083952 Integer coefficients of A(x), where 1<=a(n)<=2, such that A(x)^(1/2) consists entirely of integer coefficients. %C A083952 More generally, "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. Is this sequence periodic? %H A083952 Robert G. Wilson v, Table of n, a(n) for n = 0..5506 %H A083952 N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745. %t A083952 a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n - 1}]}, If[ IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Table[ a[n], {n, 0, 104}] (* from Robert G. Wilson v (rgwv@rgwv.com), Nov 25 2006 *) %t A083952 s = 0; a[n_] := a[n] = Block[{}, If[IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], s = s + x^n; 1, s = s + 2 x^n; 2]]; Table[ a@n, {n, 0, 104}] (* from Robert G. Wilson v (rgwv@rgwv.com), Sep 08 2007 *) %Y A083952 Cf. A084202 (A(x)^(1/2)), A108335 (A084202 mod 4), A108336 (A084202 mod 2), A108340 (a(n) mod 2). Positions of 1's: A108783. %Y A083952 Cf. A083953, A083954, A083945, A083946. %Y A083952 Adjacent sequences: A083949 A083950 A083951 this_sequence A083953 A083954 A083955 %Y A083952 Sequence in context: A029428 A101422 A070304 this_sequence A043529 A080942 A099812 %K A083952 nonn,nice %O A083952 0,2 %A A083952 Paul D. Hanna (pauldhanna(AT)juno.com), May 09 2003 %E A083952 More terms from njas, Jul 02 2005 %I A043529 %S A043529 1,2,1,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, %T A043529 1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, %U A043529 2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 %N A043529 Number of distinct base 2 digits of n. %Y A043529 Adjacent sequences: A043526 A043527 A043528 this_sequence A043530 A043531 A043532 %Y A043529 Sequence in context: A101422 A070304 A083952 this_sequence A080942 A099812 A068068 %K A043529 nonn,base %O A043529 1,2 %A A043529 Clark Kimberling (ck6(AT)evansville.edu) %I A080942 %S A080942 1,1,2,1,2,2,2,1,2,2,2,2,2,2,3,1,2,2,2,2,2,2,2,2,2,2,3,2,2,3,2,1,2,2,2, %T A080942 2,2,2,3,2,2,2,2,2,3,2,2,2,2,2,3,2,2,3,2,2,2,2,2,3,2,2,4,1,2,2,2,2,2,2, %U A080942 2,2,2,2,3,2,2,3,2,2,2,2,2,2,3,2,3,2,2,3,2,2,2,2,2,2,2,2,3,2,2,3,2,2,2 %N A080942 Number of divisors of n that are also suffices of n in binary representation. %C A080942 a(n)=1 iff n=2^k (A000079), the only divisor is n itself; %C A080942 for a(n)>1 the other trivial divisor is 1 for odd numbers and 2 for even numbers (A057716); %C A080942 a(A080943(n))=2; a(A080945(n))>2; a(A080946(n))=3; a(A080947(n))>3; %C A080942 a(n) <= A000005(n); for odd primes p: a(p)=2; %C A080942 a(A080948(n))=n and a(m)2. Christian G. Bower (bowerc(AT)usa.net) May 18, 2005. %C A068068 Number of primitive Pythagorean triangles with leg 4n. For smallest (even) leg of exactly 2^n PPTs, see A088860. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 12 2006 %D A068068 L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213. (See Table 1.) %H A068068 Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf) %H A068068 N. J. A. Sloane, Transforms %F A068068 a(n) = A034444(2n)/2. If n is even, a(n) = 2^(omega(n)-1); if n is odd, a(n) = 2^omega(n). Here omega(n) = A001221(n) is the number of distinct prime divisors of n. %F A068068 a(n)=A024361(4n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 12 2006 %t A068068 a[n_] := Length[Select[Divisors[n], OddQ[ # ]&&GCD[ #, n/# ]==1&]] %Y A068068 Cf. A056901, A068067. %Y A068068 Adjacent sequences: A068065 A068066 A068067 this_sequence A068069 A068070 A068071 %Y A068068 Sequence in context: A043529 A080942 A099812 this_sequence A092505 A066086 A103318 %K A068068 nonn,mult %O A068068 1,3 %A A068068 Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 19 2002 %E A068068 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Jun 08 2002 %I A092505 %S A092505 1,1,2,1,2,2,2,1,2,2,2,2,2,4,2,1,2,2,2,2,2,4,2,2,2,4,2,4,2,4,2,1,2,2,2, %T A092505 2,2,4,2,2,2,4,2,4,2,4,2,2,2,4,2,4,2,4,2,4,2,4,2,8,2,4,2,1,2,2,2,2,2,4, %U A092505 2,2,2,4,2,4,2,4,2,2,2,4,2,4,2,4,2,4,2,4,2,8,2,4,2,2,2,4,2,4,2,4 %N A092505 A002430(n) / A046990(n), n>0. %o A092505 (PARI) a(n)=if(n<1,0,numerator(polcoeff(Ser(tan(x)),2*n-1))/numerator(polcoeff(Ser(log(1/cos(x))),2*n))","))) %Y A092505 Adjacent sequences: A092502 A092503 A092504 this_sequence A092506 A092507 A092508 %Y A092505 Sequence in context: A080942 A099812 A068068 this_sequence A066086 A103318 A002321 %K A092505 nonn %O A092505 1,3 %A A092505 Ralf Stephan, Apr 05 2004 %I A066086 %S A066086 1,1,2,1,2,2,2,1,2,2,2,2,2,6,8,1,2,2,2,2,4,2,2,2,2,6,2,6,2,8,2,1,4,2, %T A066086 24,2,2,6,8,2,2,12,2,2,8,2,2,2,2,2,8,6,2,2,8,6,4,2,2,8,2,6,4,1,12,4,2, %U A066086 2,4,24,2,2,2,6,8,6,12,24,2,2,2,2,2,12,4,6,8,2,2,8,8,2,4,2,24,2,2,6,4 %N A066086 GCD[A048250(n), A023900(n)] = GCD[Sigma[A007947(n)],Phi[A007947(n)]] = GCD[A000203[A007947(n)], A000010[A007947(n)]]. %e A066086 Frequently equals A009223 (i.e. GCD of sigma and phi of n), but A066086 and A009223 are not identical. %e A066086 n=12: GCD[sigma[n],phi[n]]>GCD[sigma[core(n)],Phi[core(n)]], %t A066086 ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Table[g2[w], {w, 1, 128}] %Y A066086 Cf. A048250, A023900, A000203, A007947, A000010, A009223, A066087. %Y A066086 Adjacent sequences: A066083 A066084 A066085 this_sequence A066087 A066088 A066089 %Y A066086 Sequence in context: A099812 A068068 A092505 this_sequence A103318 A002321 A043530 %K A066086 nonn %O A066086 1,3 %A A066086 Labos E. (labos(AT)ana.sote.hu), Dec 04 2001 %I A103318 %S A103318 1,1,2,1,2,2,2,1,2,2,3,1,2,2,2,1,2,2,3,2,2,2,2,1,2,2,3,1,2,2,2,1,2, %T A103318 2,3,2,3,2,2,1,2,2,3,1,2,2,2,1,2,2,3,2,2,2,2,1,2,2,3,1,2,2,2,1,2,2, %U A103318 3,2,3,3,2,1,2,2,3,1,2,2,2,1,2,2,3,2,2,2,2,1,2,2,3,1,2,2,2,1,2,2,3 %N A103318 Number of solutions i in range [0,n-1] to i == 0 mod 2^(n-i). %C A103318 i=0 is always a solution. %C A103318 a(n) is the number of 1's in (A103745(n) written in base 2). - Philippe DELEHAM, Apr 02 2005 %H A103318 David Applegate, Benoit Cloitre, Philippe DELEHAM and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps]. %F A103318 a(n) = A104234(2^n - n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Apr 21 2005 %e A103318 For n = 11 solutions are i = 0, 8 and 10. Four solutions occur for the first time at n = 2059: they are i = 0, 2048, 2056, 2058. Five solutions occur for the first time at n = 2^2059 + 2059 (see A034797). %p A103318 f:= proc (n) local t1, l; t1 := 0; for l to n do if `mod`(n-l,2^l) = 0 then t1 := t1+1 end if end do; t1 end proc; %t A103318 f[n_] := Block[{c = 1, k = Max[1, n - Floor[ Log[2, n] + 2]]}, While[k < n, If[ Mod[k, 2^(n - k)] == 0, c++ ]; k++ ]; c]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Mar 21 2005) %Y A103318 For records see A034797. Cf. A103745. %Y A103318 Adjacent sequences: A103315 A103316 A103317 this_sequence A103319 A103320 A103321 %Y A103318 Sequence in context: A068068 A092505 A066086 this_sequence A002321 A043530 A055718 %K A103318 nonn %O A103318 1,3 %A A103318 njas, Mar 21 2005 %I A002321 M0102 N0038 %S A002321 1,0,1,1,2,1,2,2,2,1,2,2,3,2,1,1,2,2,3,3,2,1,2,2,2,1,1, %T A002321 1,2,3,4,4,3,2,1,1,2,1,0,0,1,2,3,3,3,2,3,3,3,3,2,2,3,3, %U A002321 2,2,1,0,1,1,2,1,1,1,0,1,2,2,1,2,3,3,4,3,3,3,2,3,4,4,4 %V A002321 1,0,-1,-1,-2,-1,-2,-2,-2,-1,-2,-2,-3,-2,-1,-1,-2,-2,-3,-3,-2,-1,-2,-2,-2,-1,-1, %W A002321 -1,-2,-3,-4,-4,-3,-2,-1,-1,-2,-1,0,0,-1,-2,-3,-3,-3,-2,-3,-3,-3,-3,-2,-2,-3,-3, %X A002321 -2,-2,-1,0,-1,-1,-2,-1,-1,-1,0,-1,-2,-2,-1,-2,-3,-3,-4,-3,-3,-3,-2,-3,-4,-4,-4 %N A002321 Mertens's function: Sum_{1<=k<=n} mu(k), where mu = Moebius function (A008683). %C A002321 Also determinant of n X n (0,1) matrix defined by A(i,j)=1 if j=1 or i divides j. %D A002321 J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347. %D A002321 E. Landau, Vorlesungen ueber Zahlentheorie, Chelsea, NY, Vol. 2, p. 157. %D A002321 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10. %D A002321 N. C. Ng, The summatory function of the Mobius function, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-316. %D A002321 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.1. %D A002321 R. D. von Sterneck, Empirische Untersuchung ueber den Verlauf der zahlentheoretischer Function sigma(n) = Sum_{x=1..n} mu(x) im Intervalle von 0 bis 150 000, Sitzungsbericht der Kaiserlichen Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschaftlichen Klasse, 2a, v. 106, 1897, 835-1024. %H A002321 T. D. Noe, Table of n, a(n) for n = 1..10000 %H A002321 G. J. Chaitin, [math/0306042] Thoughts on the Riemann hypothesis %H A002321 J. B. Conrey, The Riemann Hypothesis %H A002321 F. Dress, Fonction sommatoire de la fonction de Moebius. 1. Majorations experimentales. %H A002321 F. Dress, Fonction sommatoire de la fonction de Moebius. 2. Majorations asymptotiques elementaires. %H A002321 M. El-Marraki, Fonction sommatoire de la fonction mu de Moebius %H A002321 A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. reine angew. Math., 357 (1985), pp. 138-160. %H A002321 G. Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens %H A002321 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A002321 Eric Weisstein's World of Mathematics, Redheffer Matrix %H A002321 Wikipedia, Mertens function %F A002321 Assuming the Riemann hypothesis, a(n) = O(x^(1/2 + eps)) for every eps > 0 (Littlewood - see Land