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Total number of prime parts in all compositions of n.
10

%I #19 Oct 30 2020 15:03:53

%S 0,0,1,3,7,18,42,98,222,497,1100,2413,5250,11350,24398,52193,111180,

%T 235949,499074,1052502,2213710,4644833,9724492,20318637,42376578,

%U 88231765,183420748,380755932,789340736,1634339217,3379993922,6982618822,14410499598,29711523105

%N Total number of prime parts in all compositions of n.

%H Alois P. Heinz, <a href="/A102291/b102291.txt">Table of n, a(n) for n = 0..3313</a>

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

%F a(n) = Sum_{k=1..floor(n/2)} k * A224344(n,k). - _Alois P. Heinz_, Aug 06 2019

%p a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+

%p `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n))

%p end:

%p seq(a(n), n=0..33); # _Alois P. Heinz_, Aug 06 2019

%t a[n_] := a[n] = If[n==0, 0, Sum[a[n-j] + If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}]];

%t a /@ Range[0, 33] (* _Jean-François Alcover_, Oct 30 2020, after _Alois P. Heinz_ *)

%Y Cf. A037032, A224344, A336632.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Feb 19 2005

%E More terms from _Joshua Zucker_, May 10 2006