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

1,4

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A000009, A047966.

Sequence in context: A319711 A319713 A063717 * A243329 A051953 A079277

Adjacent sequences: A024991 A024992 A024993 * A024995 A024996 A024997

Keyword

nonn

Author

Clark Kimberling

Extensions

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

Status

approved