A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004

Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018

Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). - Gary W. Adamson, Jun 08 2009

Examples

			The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - _Gus Wiseman_, Apr 16 2018
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Crossrefs

Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016

Mathematica

b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)
Table[DivisorSum[n,PartitionsQ],{n,20}] (* Gus Wiseman, Apr 16 2018 *)

PARI

N = 66; q='q+O('q^N);
D(q)=eta(q^2)/eta(q); \\ A000009
Vec( sum(e=1,N,D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014

Formula

G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003

G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018

a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018