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A336632 Number of prime parts, counted without multiplicity, in all compositions of n.

%I A336632 #12 May 30 2022 08:14:58
%S A336632 0,0,1,3,6,15,33,74,160,344,731,1544,3237,6753,14022,29009,59819,
%T A336632 123010,252341,516560,1055476,2153115,4385889,8922556,18131000,
%U A336632 36805009,74643126,151255021,306267833,619719217,1253191291,2532750315,5116124712,10329574480
%N A336632 Number of prime parts, counted without multiplicity, in all compositions of n.
%H A336632 Alois P. Heinz, <a href="/A336632/b336632.txt">Table of n, a(n) for n = 0..1000</a>
%e A336632 a(4) = 0 + 1 + 1 + 1 + 1 + 1 + 1 + 0 = 6: 1111, 11(2), 1(2)1, (2)11, (2)2, 1(3), (3)1, 4.
%p A336632 b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
%p A336632       `if`(i<1, 0, add((h-> [0, `if`(j>0 and isprime(i),
%p A336632        h[1], 0)]+h)(b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
%p A336632     end:
%p A336632 a:= n-> b(n$2, 0)[2]:
%p A336632 seq(a(n), n=0..38);
%t A336632 b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
%t A336632      If[i < 1, 0, Sum[Function[h, {0, If[j > 0 && PrimeQ[i],
%t A336632      h[[1]], 0]} + h][b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
%t A336632 a[n_] := b[n, n, 0][[2]];
%t A336632 Table[a[n], {n, 0, 38}] (* _Jean-François Alcover_, May 30 2022, after _Alois P. Heinz_ *)
%Y A336632 Cf. A000040, A102291, A336579.
%K A336632 nonn
%O A336632 0,4
%A A336632 _Alois P. Heinz_, Jul 28 2020