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

0, 0, 1, 3, 6, 15, 33, 74, 160, 344, 731, 1544, 3237, 6753, 14022, 29009, 59819, 123010, 252341, 516560, 1055476, 2153115, 4385889, 8922556, 18131000, 36805009, 74643126, 151255021, 306267833, 619719217, 1253191291, 2532750315, 5116124712, 10329574480

Offset: 0

Author

Alois P. Heinz, Jul 28 2020

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000040, A102291, A336579.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
      `if`(i<1, 0, add((h-> [0, `if`(j>0 and isprime(i),
       h[1], 0)]+h)(b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
     If[i < 1, 0, Sum[Function[h, {0, If[j > 0 && PrimeQ[i],
     h[[1]], 0]} + h][b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)