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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A073371 A113224 A178945 * A026571 A100099 A164267

Adjacent sequences: A309558 A309559 A309560 * A309562 A309563 A309564

Keyword

nonn

Author

Alois P. Heinz, Aug 07 2019

Status

approved