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%I #4 Feb 05 2017 13:17:31
%S 0,2,5,13,23,41,69,119,185,283,425,625,903,1285,1799,2517,3450,4699,
%T 6340,8490,11264,14870,19485,25390,32897,42395,54372,69408,88210,
%U 111612,140717,176738,221135,275776,342790,424743,524765,646420,794109,972967,1189105,1449577,1763097,2139394,2590349,3129633,3773546,4540645
%N Expansion of Sum_{p prime, i>=1} p^i*x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).
%C Total sum of prime power parts (1 excluded) in all partitions of n.
%C Convolution of the sequences A000041 and A023889.
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Sum_{p prime, i>=1} p^i*x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).
%e a(5) = 23 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 5 + 4 + 3 + 2 + 3 + 2 + 2 + 2 = 23.
%t nmax = 48; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[i]] i x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]
%Y Cf. A000041, A023889, A066186, A073118, A246655.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Feb 01 2017
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