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

0,11

Links

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A023895, A096258, A000586, A000009.

Sequence in context: A349463 A114325 A101048 * A070102 A029182 A035373

Adjacent sequences: A204386 A204387 A204388 * A204390 A204391 A204392

Keyword

nonn

Author

Reinhard Zumkeller, Jan 15 2012

Status

approved