A168656 Number of partitions of n such that the smallest part is divisible by the number of parts.

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, 53, 58, 66, 74, 83, 92, 103, 116, 130, 144, 160, 179, 199, 219, 243, 269, 298, 328, 362, 399, 441, 484, 533, 586, 645, 708, 778, 854, 937, 1026, 1124, 1230, 1347, 1470, 1607, 1756, 1917, 2089

Offset: 1

Author

Vladeta Jovovic, Dec 01 2009, Dec 04 2009

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

PARI

N=100; x='x+O('x^N);
Vec( sum(k=1,sqrtint(N), x^(k^2)/(1-x^(k^2)) / prod(i=1,k-1, 1-x^i) ) )

Formula

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

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