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%I #7 Apr 11 2017 17:02:21
%S 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,
%T 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,
%U 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
%N Number of partitions of n into odd divisors of n.
%C a(2*n+1) = A018818(2*n+1), a(A005408(n))=A018818(A005408(n));
%C a(2^k) = 1, a(A000079(n))=1;
%C for odd primes p: a(p*2^k) = 2^k + 1,
%C especially for n>1: a(A000040(n))=2, a(A100484(n))=3, a(A001749(n))=5.
%H Alois P. Heinz, <a href="/A171565/b171565.txt">Table of n, a(n) for n = 0..10000</a>
%F 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.
%p with(numtheory):
%p a:= proc(n) option remember; local b, l; l, b:= sort(
%p [select(x-> is(x:: odd), divisors(n))[]]),
%p proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p end; b(n, nops(l))
%p end:
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Mar 30 2017
%t 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]]];
%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 11 2017, after _Alois P. Heinz_ *)
%Y Cf. A038550, A065091.
%K nonn
%O 0,4
%A _Reinhard Zumkeller_, Dec 11 2009
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