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 A096276 #51 May 27 2024 22:42:16
%S A096276 0,1,2,3,5,6,8,9,12,14,16,17,21,22,24,26,31,32,36,37,41,43,45,46,53,
%T A096276 55,57,60,64,65,70,71,78,80,82,84,93,94,96,98,105,106,111,112,116,120,
%U A096276 122,123,135,137,141,143,147,148,155,157,164,166,168,169,180,181,183,187
%N A096276 Number of partitions of n with a product <=n.
%C A096276 The Heinz numbers of these partitions are given by A325044. - _Gus Wiseman_, Mar 27 2019
%D A096276 G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 198, exercise 9 (in the third edition 2015, p. 296, exercise 211).
%H A096276 Alois P. Heinz, <a href="/A096276/b096276.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H A096276 R. E. Canfield, P. Erdős and C. Pomerance, <a href="https://doi.org/10.1016/0022-314X(83)90002-1">On a Problem of Oppenheim concerning "Factorisatio Numerorum"</a>, J. Number Theory 17 (1983), 1-28.
%H A096276 Pankaj Jyoti Mahanta, <a href="https://arxiv.org/abs/2010.07353">On the number of partitions of n whose product of the summands is at most n</a>, arXiv:2010.07353 [math.CO], 2020.
%H A096276 A. Oppenheim, <a href="https://doi.org/10.1112/jlms/s1-1.4.205">On an Arithmetic Function</a>, Journal of the London Mathematical Society, 1926, 10, Vol. s1-1, Iss. 4, 205-211.
%H A096276 A. Oppenheim, <a href="https://doi.org/10.1112/jlms/s1-2.2.123">On an Arithmetic Function (II)</a>, Journal of the London Mathematical Society, 1927, 04, Vol. s1-2, Iss. 2, 123-130.
%H A096276 Csaba Sándor and Maciej Zakarczemny, <a href="https://arxiv.org/abs/2405.11600">Equal Sum and Product Problem III</a>, arXiv:2405.11600 [math.NT], 2024.
%F A096276 For n>1, a(n) = a(n-1)+1 iff n is prime.
%F A096276 Partial sums of A001055. - _Vladeta Jovovic_, Jun 24 2004
%F A096276 a(n) ~ n * exp(2*sqrt(log(n))) / (2*sqrt(Pi) * (log(n))^(3/4)) [Oppenheim, 1927]. - _Vaclav Kotesovec_, May 23 2020
%e A096276 a(6)=8 as we can have 6, 51, 411, 321, 3111, 2211, 21111, 111111, rejecting 42, 33 and 222.
%e A096276 From _Gus Wiseman_, Mar 27 2019: (Start)
%e A096276 The a(1) = 1 through a(8) = 12 partitions:
%e A096276 (1) (2) (3) (4) (5) (6) (7) (8)
%e A096276 (11) (21) (22) (41) (51) (61) (71)
%e A096276 (111) (31) (221) (321) (511) (611)
%e A096276 (211) (311) (411) (3211) (4211)
%e A096276 (1111) (2111) (2211) (4111) (5111)
%e A096276 (11111) (3111) (22111) (22211)
%e A096276 (21111) (31111) (32111)
%e A096276 (111111) (211111) (41111)
%e A096276 (1111111) (221111)
%e A096276 (311111)
%e A096276 (2111111)
%e A096276 (11111111)
%e A096276 (End)
%p A096276 g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
%p A096276 `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
%p A096276 d=numtheory[divisors](n) minus {1, n}))
%p A096276 end:
%p A096276 a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+g(n$2)) end:
%p A096276 seq(a(n), n=0..100); # _Alois P. Heinz_, Feb 26 2023
%t A096276 c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{ds, i}, ds = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; Join[{0}, Accumulate[Array[a, 100]]] (* using program from A001055, _T. D. Noe_, Apr 11 2011 *)
%t A096276 Table[Length[Select[IntegerPartitions[n],Times@@#<=n&]],{n,0,20}] (* _Gus Wiseman_, Mar 27 2019 *)
%o A096276 (PARI) { bla(n,m,v,z)=v=concat(v,m); if(!n,x=prod(k=1,length(v),v[k]); if (x<=z,c++), for(i=1,min(m,n),bla(n-i,i,v,z))); }
%o A096276 q(n)=c=0;for(i=1,n,bla(n-i,i,[],n));print1(c, ", ");
%o A096276 for(i=0,40,q(i))
%Y A096276 Cf. A001055, A028422, A114324, A301987, A319000, A319005, A319916, A325044.
%K A096276 nonn
%O A096276 0,3
%A A096276 _Jon Perry_, Jun 23 2004
%E A096276 More terms from _Vladeta Jovovic_, Jun 24 2004