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

%I

%S 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 82,94,106,122,138,157,178,201,226,254,286,321,360,402,448,501,558,

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

%C Total number of parts in all partitions of n into odd primes.

%C Convolution of A005087 and A099773.

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

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

%e a(14) = 8 because we have [11, 3], [7, 7], [5, 3, 3, 3] and 2 + 2 + 4 = 8.

%t 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 (PARI)

%o 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 {my(n=60); Vec(sumparts(n, v->v>2 && isprime(v)), -n)} \\ _Andrew Howroyd_, Dec 28 2017

%Y Cf. A005087, A065091, A084993, A099773.

%K nonn

%O 1,6

%A _Ilya Gutkovskiy_, Jan 23 2017

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Last modified August 21 18:26 EDT 2019. Contains 326168 sequences. (Running on oeis4.)