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A281544 Expansion of Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).

%I A281544 #7 Dec 28 2017 21:31:26
%S A281544 0,0,1,0,1,2,1,2,3,4,4,6,7,8,11,12,15,18,20,26,29,34,40,46,54,62,71,
%T A281544 82,94,106,122,138,157,178,201,226,254,286,321,360,402,448,501,558,
%U A281544 619,690,764,846,938,1036,1145,1264,1392,1532,1687,1854,2036,2234,2448,2680,2934,3210,3507,3828,4178,4554,4961,5404
%N A281544 Expansion of Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).
%C A281544 Total number of parts in all partitions of n into odd primes.
%C A281544 Convolution of A005087 and A099773.
%H A281544 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A281544 G.f.: Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).
%e A281544 a(14) = 8 because we have [11, 3], [7, 7], [5, 3, 3, 3] and 2 + 2 + 4 = 8.
%t A281544 nmax = 68; 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 A281544 (PARI)
%o A281544 sumparts(n, pred)={sum(k=1, n, 1/(1-pred(k)*x^k) - 1 + O(x*x^n))/prod(k=1, n, 1-pred(k)*x^k + O(x*x^n))}
%o A281544 {my(n=60); Vec(sumparts(n, v->v>2 && isprime(v)), -n)} \\ _Andrew Howroyd_, Dec 28 2017
%Y A281544 Cf. A005087, A065091, A084993, A099773.
%K A281544 nonn
%O A281544 1,6
%A A281544 _Ilya Gutkovskiy_, Jan 23 2017