A281617 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).

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, 35, 69, 49, 80, 68, 110, 83, 137, 112, 166, 150, 215, 178, 268, 239, 324, 303, 406, 365, 504, 472, 604, 584, 747, 708, 917, 888, 1089, 1085, 1337, 1311, 1618, 1606, 1916, 1954, 2332, 2334, 2782, 2829, 3300, 3407, 3963

Offset: 0

Author

Ilya Gutkovskiy, Jan 25 2017

Keywords

• nonn

Comments

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

Convolution of A086971 and A101048.

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

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

Programs

Maple

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0$2],
     `if`(i>n, 0, (p-> p+[0, p[1]])(b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
    end:
a:= n-> b(n, h(n))[2]:
seq(a(n), n=0..70);  # Alois P. Heinz, May 19 2021

Mathematica

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]]

Formula

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).

a(n) = Sum_{k>0} k * A344447(n,k). - Alois P. Heinz, May 19 2021