A204389 Number of partitions of n into distinct composite parts.

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 2, 3, 1, 5, 3, 5, 4, 7, 4, 9, 7, 10, 9, 13, 10, 17, 14, 18, 18, 25, 22, 30, 27, 34, 36, 44, 40, 53, 52, 62, 65, 76, 74, 93, 95, 107, 113, 131, 133, 158, 164, 182, 195, 221, 229, 264, 276, 304, 329, 367, 383, 431

Offset: 0

Author

Reinhard Zumkeller, Jan 15 2012

Keywords

• nonn

Examples

			a(10) = #{10, 6+4} = 2;
a(11) = #{ } = 0;
a(12) = #{12, 8+4} = 2;
a(13) = #{9+4} = 1;
a(14) = #{14, 10+4, 8+6} = 3;
a(15) = #{15, 9+6} = 2;
a(16) = #{16, 12+4, 10+6} = 3;
a(17) = #{9+8} = 1;
a(18) = #{18, 14+4, 12+6, 10+8, 8+6+4} = 5;
a(19) = #{15+4, 10+9, 9+6+4} = 3;
a(20) = #{20, 16+4, 14+6, 12+8, 10+6+4} = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..250 from Reinhard Zumkeller)

Crossrefs

Cf. A023895, A096258, A000586, A000009.

Programs

Haskell

a204389 = p a002808_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
       b(n, i-1)+ `if`(i>n or isprime(i), 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, May 29 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i>n || PrimeQ[i], 0, b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)

Formula

G.f.: (1/(1 + x))*Product_{k>=1} (1 + x^k)/(1 + x^prime(k)). - Ilya Gutkovskiy, Dec 31 2016

G.f.: product_(i>=1) (1+x^A002808(i)). - R. J. Mathar, Mar 01 2023