A102291 Total number of prime parts in all compositions of n.

0, 0, 1, 3, 7, 18, 42, 98, 222, 497, 1100, 2413, 5250, 11350, 24398, 52193, 111180, 235949, 499074, 1052502, 2213710, 4644833, 9724492, 20318637, 42376578, 88231765, 183420748, 380755932, 789340736, 1634339217, 3379993922, 6982618822, 14410499598, 29711523105

Offset: 0

Author

Vladeta Jovovic, Feb 19 2005

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3313

Crossrefs

Cf. A037032, A224344, A336632.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 06 2019

Mathematica

a[n_] := a[n] = If[n==0, 0, Sum[a[n-j] + If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}]];
a /@ Range[0, 33] (* Jean-François Alcover, Oct 30 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=1} x^prime(k)*(1-x)^2/(1-2*x)^2.

a(n) = Sum_{k=1..floor(n/2)} k * A224344(n,k). - Alois P. Heinz, Aug 06 2019

Extensions

More terms from Joshua Zucker, May 10 2006