A014648 Number of partitions of n into its divisors with at least one part of size 1.

1, 1, 1, 2, 1, 5, 1, 6, 3, 8, 1, 33, 1, 11, 11, 26, 1, 66, 1, 76, 15, 17, 1, 378, 5, 20, 18, 136, 1, 648, 1, 166, 23, 26, 23, 1954, 1, 29, 27, 1212, 1, 1500, 1, 309, 261, 35, 1, 8600, 7, 378, 35, 422, 1, 2877, 35, 2803, 39, 44, 1, 66129, 1, 47, 461, 1626, 41, 4936

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

Cf. A018818.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local b, l; l:= sort([divisors(n)[]]):
      b:= 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; forget(b):
      b(n-1, nops(l))
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Feb 05 2014

Mathematica

a[n_] := a[n] = Module[{b, l}, l = Divisors[n]; 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-1, Length[l]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 07 2015, after Alois P. Heinz *)

Formula

Coefficient of x^(n-1) in expansion of 1/Product_{d divides n} (1-x^d). - Vladeta Jovovic, Apr 11 2004