A171565 Number of partitions of n into odd divisors of n.

1, 1, 1, 2, 1, 2, 3, 2, 1, 5, 3, 2, 5, 2, 3, 14, 1, 2, 12, 2, 5, 18, 3, 2, 9, 7, 3, 23, 5, 2, 54, 2, 1, 26, 3, 26, 35, 2, 3, 30, 9, 2, 72, 2, 5, 286, 3, 2, 17, 9, 18, 38, 5, 2, 93, 38, 9, 42, 3, 2, 275, 2, 3, 493, 1, 44, 108, 2, 5, 50, 110, 2, 117, 2, 3, 698, 5, 50, 126, 2, 17, 239, 3, 2, 375, 56

Offset: 0

Author

Reinhard Zumkeller, Dec 11 2009

Keywords

• nonn

Comments

a(2*n+1) = A018818(2*n+1), a(A005408(n))=A018818(A005408(n));

a(2^k) = 1, a(A000079(n))=1;

for odd primes p: a(p*2^k) = 2^k + 1,

especially for n>1: a(A000040(n))=2, a(A100484(n))=3, a(A001749(n))=5.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A038550, A065091.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> is(x:: odd), divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

Mathematica

a[0] = 1; a[n_] := a[n] = Module[{b, l}, l = Select[Divisors[n], OddQ]; b[m_, i_] := b[m, i] = If[m == 0, 1, If[i < 1, 0, b[m, i-1] + If[l[[i]] > m, 0, b[m - l[[i]], i]]]]; b[n, Length[l]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 11 2017, after Alois P. Heinz *)

Formula

a(n) = f(n,n,1) with f(n,m,k) = if k<=m then f(n,m,k+2)+f(n,m-k,k)*0^(n mod k) else 0^m.