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A281545 Expansion of Sum_{k>=2} x^prime(k)/(1 + x^prime(k)) * Product_{k>=2} (1 + x^prime(k)).

%I A281545 #11 Dec 30 2017 09:49:48
%S A281545 0,0,1,0,1,0,1,2,0,2,1,2,1,2,3,4,1,4,4,4,6,4,7,6,6,8,9,8,10,6,13,12,
%T A281545 12,14,15,16,16,18,23,22,19,24,24,30,28,30,33,34,34,40,44,46,44,46,58,
%U A281545 56,60,64,65,68,70,80,86,88,87,94,101,112,114,116,125,130,132,148,159,162,163,168,190,196
%N A281545 Expansion of Sum_{k>=2} x^prime(k)/(1 + x^prime(k)) * Product_{k>=2} (1 + x^prime(k)).
%C A281545 Total number of parts in all partitions of n into distinct odd primes.
%H A281545 Andrew Howroyd, <a href="/A281545/b281545.txt">Table of n, a(n) for n = 1..1000</a>
%H A281545 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A281545 G.f.: Sum_{k>=2} x^prime(k)/(1 + x^prime(k)) * Product_{k>=2} (1 + x^prime(k)).
%e A281545 a(23) = 7 because we have [23], [13, 7, 3], [11, 7, 5] and 1 + 3 + 3 = 7.
%t A281545 nmax = 80; Rest[CoefficientList[Series[Sum[x^Prime[k]/(1 + x^Prime[k]), {k, 2, nmax}] Product[1 + x^Prime[k], {k, 2, nmax}], {x, 0, nmax}], x]]
%o A281545 (PARI)
%o A281545 sumparts(n, pred)={sum(k=1, n, 1 - 1/(1+pred(k)*x^k) + O(x*x^n))*prod(k=1, n, 1+pred(k)*x^k + O(x*x^n))}
%o A281545 {my(n=60); Vec(sumparts(n, v->v>2 && isprime(v)), -n)} \\ _Andrew Howroyd_, Dec 28 2017
%Y A281545 Cf. A024938, A024939, A065091.
%K A281545 nonn
%O A281545 1,8
%A A281545 _Ilya Gutkovskiy_, Jan 23 2017