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

1,4

Links

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

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

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 *)

Prog

(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) ) )

Crossrefs

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

Sequence in context: A241086 A194818 A029081 * A005862 A293254 A046052

Adjacent sequences: A168653 A168654 A168655 * A168657 A168658 A168659

Keyword

easy,nonn,changed

Author

Vladeta Jovovic, Dec 01 2009, Dec 04 2009

Status

approved