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

0,4

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

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.

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 *)

Crossrefs

Cf. A038550, A065091.

Sequence in context: A339010 A332842 A352696 * A328266 A359895 A115116

Adjacent sequences: A171562 A171563 A171564 * A171566 A171567 A171568

Keyword

nonn

Author

Reinhard Zumkeller, Dec 11 2009

Status

approved