%I #6 Apr 28 2018 17:13:59

%S 1,1,2,3,4,7,9,15,21,31,47,67,102,148,220,325,477,709,1041,1542,2274,

%T 3355,4959,7311,10804,15940,23535,34747,51281,75723,111762,165005,

%U 243578,359567,530831,783585,1156799,1707662,2520913,3721467,5493674,8110012,11972133,17673686

%N Expansion of 1/((1 - x)*(1 - Sum_{k>=1} x^prime(k))).

%C Partial sums of A023360.

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%p b:= proc(n) option remember; `if`(n=0, 1,

%p add(`if`(isprime(j), b(n-j), 0), j=2..n))

%p end:

%p a:= proc(n) option remember;

%p `if`(n<0, 0, b(n)+a(n-1))

%p end:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 28 2018

%t nmax = 43; CoefficientList[Series[1/((1 - x) (1 - Sum[x^Prime[k], {k, 1, nmax}])), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[Boole[PrimeQ[k]] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 43}]]

%Y Cf. A000040, A010051, A023360, A034891.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 28 2018