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

1,2

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

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A236129 A240219 A028298 * A360243 A360683 A317085

Adjacent sequences: A047963 A047964 A047965 * A047967 A047968 A047969

Keyword

nonn

Author

N. J. A. Sloane

Status

approved