A309561 Total sum of prime parts in all compositions of n.

0, 0, 2, 7, 16, 44, 102, 244, 554, 1247, 2772, 6111, 13334, 28916, 62302, 133557, 285020, 605869, 1283362, 2710008, 5706546, 11986171, 25118500, 52529339, 109643310, 228455907, 475250388, 987177924, 2047710144, 4242128909, 8777675002, 18142184432, 37458037658

Offset: 0

Author

Alois P. Heinz, Aug 07 2019

Keywords

• nonn

Examples

			a(4) = 16: 1111, (2)11, 1(2)1, 11(2), (2)(2), (3)1, 1(3), 4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3312

Crossrefs

Cf. A000040, A001787, A073118, A102291, A336579.

Programs

Maple

a:= proc(n) option remember; add(a(n-j) +j*
      `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);

Mathematica

a[n_] := a[n] = Sum[a[n-j]+j*If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}];
a /@ Range[0, 33] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

Formula

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

a(n) ~ c * 2^n * n, where c = 0.27326606442562679135064648817419092073886899135... - Vaclav Kotesovec, Aug 18 2019