A281612 - OEIS

%I #9 Jan 27 2017 13:05:59

%S 0,0,0,1,1,3,4,8,12,20,28,45,62,92,127,181,244,340,452,614,809,1077,

%T 1401,1841,2371,3071,3923,5026,6363,8078,10149,12769,15939,19899,

%U 24676,30604,37726,46489,57007,69849,85211,103871,126119,152987,184955,223349,268898,323384,387830,464587,555168,662619,789084

%N Expansion of Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).

%C Total number of semiprime parts (A001358) in all partitions of n.

%C Convolution of A000041 and A086971.

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

%F G.f.: Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).

%e a(6) = 3 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1] and 1 + 0 + 1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 3.

%t nmax = 53; Rest[CoefficientList[Series[Sum[Floor[PrimeOmega[i]/2] Floor[2/PrimeOmega[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

%Y Cf. A000041, A001358, A037032, A073335, A086971, A281573.

%K nonn

%O 1,6

%A _Ilya Gutkovskiy_, Jan 25 2017