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

%I A309561 #15 Jan 06 2021 12:31:32
%S A309561 0,0,2,7,16,44,102,244,554,1247,2772,6111,13334,28916,62302,133557,
%T A309561 285020,605869,1283362,2710008,5706546,11986171,25118500,52529339,
%U A309561 109643310,228455907,475250388,987177924,2047710144,4242128909,8777675002,18142184432,37458037658
%N A309561 Total sum of prime parts in all compositions of n.
%H A309561 Alois P. Heinz, <a href="/A309561/b309561.txt">Table of n, a(n) for n = 0..3312</a>
%F A309561 G.f.: Sum_{k>=1} prime(k)*x^prime(k)*(1-x)^2/(1-2*x)^2.
%F A309561 a(n) ~ c * 2^n * n, where c = 0.27326606442562679135064648817419092073886899135... - _Vaclav Kotesovec_, Aug 18 2019
%e A309561 a(4) = 16: 1111, (2)11, 1(2)1, 11(2), (2)(2), (3)1, 1(3), 4.
%p A309561 a:= proc(n) option remember; add(a(n-j) +j*
%p A309561       `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n)
%p A309561     end:
%p A309561 seq(a(n), n=0..33);
%t A309561 a[n_] := a[n] = Sum[a[n-j]+j*If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}];
%t A309561 a /@ Range[0, 33] (* _Jean-François Alcover_, Jan 03 2021, after _Alois P. Heinz_ *)
%Y A309561 Cf. A000040, A001787, A073118, A102291, A336579.
%K A309561 nonn
%O A309561 0,3
%A A309561 _Alois P. Heinz_, Aug 07 2019