A168656 - OEIS

%I #23 Dec 22 2024 10:59:19

%S 1,1,1,2,2,2,2,3,4,4,5,6,7,7,8,10,11,13,15,18,20,23,25,29,33,36,41,47,

%T 53,58,66,74,83,92,103,116,130,144,160,179,199,219,243,269,298,328,

%U 362,399,441,484,533,586,645,708,778,854,937,1026,1124,1230,1347,1470,1607,1756,1917,2089

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

%H Seiichi Manyama, <a href="/A168656/b168656.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} x^(k^2)/((1-x^(k^2)) * Product_{i=1..k-1} (1-x^i)).

%F a(n) ~ c * exp(2*Pi*sqrt(n/15)) / n^(3/4), where c = 1 / (2 * 3^(1/4) * sqrt(5) * phi^(3/2)) = 0.08255116908... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 17 2024

%t nmax = 100; Rest[CoefficientList[Series[Sum[x^(k^2)/((1 - x^(k^2))*Product[1 - x^j, {j, 1, k-1}]), {k, 1, Sqrt[nmax]}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Oct 16 2024 *)

%t Table[Count[IntegerPartitions[n],_?(Mod[#[[-1]],Length[#]]==0&)],{n,70}] (* _Harvey P. Dale_, Dec 22 2024 *)

%o (PARI)

%o N=100; x='x+O('x^N);

%o Vec( sum(k=1,sqrtint(N), x^(k^2)/(1-x^(k^2)) / prod(i=1,k-1, 1-x^i) ) )

%Y Cf. A000041, A006141, A073336, A079501, A168655, A168657, A168659.

%K easy,nonn,changed

%O 1,4

%A _Vladeta Jovovic_, Dec 01 2009, Dec 04 2009