A024994 Number of periodic partitions of n: each part occurs more than once and the same number of times.

0, 1, 1, 2, 1, 4, 1, 4, 3, 5, 1, 10, 1, 7, 6, 10, 1, 16, 1, 17, 8, 14, 1, 31, 4, 20, 11, 31, 1, 48, 1, 42, 15, 40, 9, 79, 1, 56, 21, 87, 1, 111, 1, 105, 41, 106, 1, 185, 6, 157, 41, 187, 1, 254, 16, 259, 57, 258, 1, 425, 1, 342, 92, 432, 22, 557, 1, 554, 107, 627, 1, 875, 1, 762, 175, 922, 18, 1173

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			E.g. 6 = 1+1+1+1+1+1 = 2+2+2 = 3+3 = 2+1+2+1, so a(6)=4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000009, A047966.

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) minus {n}):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* b[n-j], {j, 1, n}]/n]; a[n_] := Sum[b[d], {d, Divisors[n] ~Complement~ {n}}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 25 2017, after Alois P. Heinz *)

Formula

a(n) = Sum(q(k)), where k divides n, k < n, where q(n) = A000009(n), distinct partitions. - Alford Arnold

Extensions

a(1) set to 0 by Alois P. Heinz, Jul 11 2016