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A204389 Number of partitions of n into distinct composite parts.

%I A204389 #21 Oct 27 2023 20:00:35
%S A204389 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,
%T A204389 18,18,25,22,30,27,34,36,44,40,53,52,62,65,76,74,93,95,107,113,131,
%U A204389 133,158,164,182,195,221,229,264,276,304,329,367,383,431
%N A204389 Number of partitions of n into distinct composite parts.
%H A204389 Alois P. Heinz, <a href="/A204389/b204389.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 0..250 from Reinhard Zumkeller)
%F A204389 G.f.: (1/(1 + x))*Product_{k>=1} (1 + x^k)/(1 + x^prime(k)). - _Ilya Gutkovskiy_, Dec 31 2016
%F A204389 G.f.: product_(i>=1) (1+x^A002808(i)). - _R. J. Mathar_, Mar 01 2023
%e A204389 a(10) = #{10, 6+4} = 2;
%e A204389 a(11) = #{ } = 0;
%e A204389 a(12) = #{12, 8+4} = 2;
%e A204389 a(13) = #{9+4} = 1;
%e A204389 a(14) = #{14, 10+4, 8+6} = 3;
%e A204389 a(15) = #{15, 9+6} = 2;
%e A204389 a(16) = #{16, 12+4, 10+6} = 3;
%e A204389 a(17) = #{9+8} = 1;
%e A204389 a(18) = #{18, 14+4, 12+6, 10+8, 8+6+4} = 5;
%e A204389 a(19) = #{15+4, 10+9, 9+6+4} = 3;
%e A204389 a(20) = #{20, 16+4, 14+6, 12+8, 10+6+4} = 5.
%p A204389 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
%p A204389        b(n, i-1)+ `if`(i>n or isprime(i), 0, b(n-i, i-1))))
%p A204389     end:
%p A204389 a:= n-> b(n$2):
%p A204389 seq(a(n), n=0..70);  # _Alois P. Heinz_, May 29 2013
%t A204389 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_ *)
%o A204389 (Haskell)
%o A204389 a204389 = p a002808_list where
%o A204389    p _      0 = 1
%o A204389    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
%Y A204389 Cf. A023895, A096258, A000586, A000009.
%K A204389 nonn
%O A204389 0,11
%A A204389 _Reinhard Zumkeller_, Jan 15 2012