This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A087153 #53 Jan 20 2025 12:36:58
%S A087153 1,0,1,1,1,2,3,3,5,5,8,9,13,15,20,24,30,37,47,55,71,83,103,123,151,
%T A087153 178,218,257,310,366,440,515,617,722,857,1003,1184,1380,1625,1889,
%U A087153 2214,2570,3000,3472,4042,4669,5414,6244,7221,8303,9583,10998,12655,14502
%N A087153 Number of partitions of n into nonsquares.
%C A087153 Also, number of partitions of n where there are fewer than k parts equal to k for all k. - _Jon Perry_ and _Vladeta Jovovic_, Aug 04 2004. E.g. a(8)=5 because we have 8=6+2=5+3=4+4=3+3+2.
%C A087153 Convolution of A276516 and A000041. - _Vaclav Kotesovec_, Dec 30 2016
%C A087153 From _Gus Wiseman_, Apr 02 2019: (Start)
%C A087153 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The Heinz numbers of the integer partitions described in Perry and Jovovic's comment are given by A325128, while the Heinz numbers of the integer partitions described in the name are given by A325129. In the former case, the first 10 terms count the following integer partitions:
%C A087153 () (2) (3) (4) (5) (6) (7) (8) (9)
%C A087153 (32) (33) (43) (44) (54)
%C A087153 (42) (52) (53) (63)
%C A087153 (62) (72)
%C A087153 (332) (432)
%C A087153 while in the latter case they count the following:
%C A087153 () (2) (3) (22) (5) (6) (7) (8) (63)
%C A087153 (32) (33) (52) (53) (72)
%C A087153 (222) (322) (62) (333)
%C A087153 (332) (522)
%C A087153 (2222) (3222)
%C A087153 (End)
%D A087153 G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. See page 48.
%H A087153 T. D. Noe and Vaclav Kotesovec, <a href="/A087153/b087153.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A087153 Daniel I. A. Cohen, <a href="http://dx.doi.org/10.1016/0097-3165(81)90057-1">PIE-sums: a combinatorial tool for partition theory</a>. J. Combin. Theory Ser. A 31 (1981), no. 3, 223--236. MR0635367 (82m:10026). See Cor. 5. - _N. J. A. Sloane_, Mar 27 2012
%H A087153 James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Sellers/sellers58.html">Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%F A087153 G.f.: Product_{m>0} (1-x^(m^2))/(1-x^m). - _Vladeta Jovovic_, Aug 21 2003
%F A087153 a(n) = (1/n)*Sum_{k=1..n} (A000203(k)-A035316(k))*a(n-k), a(0)=1. - _Vladeta Jovovic_, Aug 21 2003
%F A087153 G.f.: Product_{i>=1} (Sum_{j=0..i-1} x^(i*j)). - _Jon Perry_, Jul 26 2004
%F A087153 a(n) ~ exp(Pi*sqrt(2*n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / 2^(3/4) - 3*Zeta(3/2)^2/(32*Pi)) * sqrt(Pi) / (2^(3/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Dec 30 2016
%e A087153 n=7: 2+5 = 2+2+3 = 7: a(7)=3;
%e A087153 n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;
%e A087153 n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.
%p A087153 g:=product((1-x^(i^2))/(1-x^i),i=1..70):gser:=series(g,x=0,60):seq(coeff(gser,x^n),n=1..53); # _Emeric Deutsch_, Feb 09 2006
%t A087153 nn=54; CoefficientList[ Series[ Product[ Sum[x^(i*j), {j, 0, i - 1}], {i, 1, nn}], {x, 0, nn}], x] (* _Robert G. Wilson v_, Aug 05 2004 *)
%t A087153 nmax = 100; CoefficientList[Series[Product[(1 - x^(k^2))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 29 2016 *)
%o A087153 (Haskell)
%o A087153 a087153 = p a000037_list where
%o A087153 p _ 0 = 1
%o A087153 p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A087153 -- _Reinhard Zumkeller_, Apr 25 2013
%o A087153 (PARI) first(n)=my(x='x+O('x^(n+1))); Vec(prod(m=1,sqrtint(n), (1-x^m^2)/(1-x^m))*prod(m=sqrtint(n)+1,n,1/(1-x^m))) \\ _Charles R Greathouse IV_, Aug 28 2016
%Y A087153 Cf. A087154, A001156, A000009, A000037, A052335 (<=k parts of k).
%Y A087153 Cf. A115584, A172151, A225044, A264393, A276516.
%Y A087153 Cf. A033461, A114639, A117144, A276429, A324572, A324588, A325128, A325129.
%K A087153 nonn
%O A087153 0,6
%A A087153 _Reinhard Zumkeller_, Aug 21 2003
%E A087153 Zeroth term added by _Franklin T. Adams-Watters_, Jan 25 2010