cp's OEIS Frontend

A168657 Number of partitions of n such that the number of parts is divisible by the smallest part.

%I A168657 #33 Jul 16 2025 13:09:12
%S A168657 1,1,2,4,6,8,12,17,25,34,48,64,87,114,151,198,258,332,428,546,695,879,
%T A168657 1108,1388,1737,2159,2680,3312,4082,5009,6138,7492,9126,11081,13429,
%U A168657 16228,19575,23547,28277,33879,40520,48354,57615,68509,81337,96388,114055
%N A168657 Number of partitions of n such that the number of parts is divisible by the smallest part.
%H A168657 Vaclav Kotesovec, <a href="/A168657/b168657.txt">Table of n, a(n) for n = 1..9464</a> (terms 1..1000 from Alois P. Heinz)
%F A168657 G.f.: Sum_{n>=1} Sum_{d|n} x^(n*d)/Product_{k=1..n-1}(1-x^k).
%F A168657 G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j^2)/Product_{k=1..i*j-1} (1-x^k). - _Seiichi Manyama_, Jan 21 2022
%F A168657 From _Vaclav Kotesovec_, Oct 17 2024: (Start)
%F A168657 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + 13*Pi / (2^(7/2) * 3^(3/2))) / sqrt(n)).
%F A168657 A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(7/2) * n^(3/2)). (End)
%p A168657 b:= proc(n, i, t) option remember;
%p A168657       `if`(n<1, 0, `if`(i=1, 1, `if`(i<1, 0,
%p A168657       `if`(irem(n, i)=0 and irem(t+n/i, i)=0, 1, 0)+
%p A168657             add(b(n-i*j, i-1, t+j), j=0..n/i))))
%p A168657     end:
%p A168657 a:= n-> b(n, n, 0):
%p A168657 seq(a(n), n=1..60);  # _Alois P. Heinz_, May 24 2012
%t A168657 b[n_, i_, t_] := b[n, i, t] = If[n<1, 0, If[i==1, 1, If[i<1, 0, If [Mod[n, i]==0 && Mod[t+n/i, i]==0, 1, 0] + Sum[b[n-i*j, i-1, t+j], {j, 0, n/i}]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Jul 01 2015, after _Alois P. Heinz_ *)
%t A168657 Table[Count[IntegerPartitions[n],_?(Mod[Length[#],#[[-1]]]==0&)],{n,50}] (* _Harvey P. Dale_, Jul 16 2025 *)
%o A168657 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(i=1, N, sum(j=1, sqrtint(N\i), x^(i*j^2)/prod(k=1, i*j-1, 1-x^k)))) \\ _Seiichi Manyama_, Jan 21 2022
%Y A168657 Cf. A079501, A168655, A168656.
%K A168657 easy,nonn
%O A168657 1,3
%A A168657 _Vladeta Jovovic_, Dec 02 2009