A281617 - OEIS

%I #11 May 21 2021 16:19:18

%S 0,0,0,0,1,0,1,0,2,1,3,0,5,2,6,3,9,3,14,7,16,10,23,12,32,20,37,28,52,

%T 35,69,49,80,68,110,83,137,112,166,150,215,178,268,239,324,303,406,

%U 365,504,472,604,584,747,708,917,888,1089,1085,1337,1311,1618,1606,1916,1954,2332,2334,2782,2829,3300,3407,3963

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

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

%C Convolution of A086971 and A101048.

%H Alois P. Heinz, <a href="/A281617/b281617.txt">Table of n, a(n) for n = 0..10000</a>

%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 = p*q, p prime, q prime} (1 - x^j).

%F a(n) = Sum_{k>0} k * A344447(n,k). - _Alois P. Heinz_, May 19 2021

%e a(12) = 5 because we have [6, 6], [4, 4, 4] and 2 + 3 = 5.

%p h:= proc(n) option remember; `if`(n=0, 0,

%p      `if`(numtheory[bigomega](n)=2, n, h(n-1)))

%p     end:

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0$2],

%p      `if`(i>n, 0, (p-> p+[0, p[1]])(b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))

%p     end:

%p a:= n-> b(n, h(n))[2]:

%p seq(a(n), n=0..70);  # _Alois P. Heinz_, May 19 2021

%t nmax = 70; Rest[CoefficientList[Series[Sum[Floor[PrimeOmega[i]/2] Floor[2/PrimeOmega[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - Floor[PrimeOmega[j]/2] Floor[2/PrimeOmega[j]] x^j, {j, 2, nmax}], {x, 0, nmax}], x]]

%Y Cf. A001358, A084993, A086971, A101048, A344447.

%K nonn

%O 0,9

%A _Ilya Gutkovskiy_, Jan 25 2017