A281612 Expansion of Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).

0, 0, 0, 1, 1, 3, 4, 8, 12, 20, 28, 45, 62, 92, 127, 181, 244, 340, 452, 614, 809, 1077, 1401, 1841, 2371, 3071, 3923, 5026, 6363, 8078, 10149, 12769, 15939, 19899, 24676, 30604, 37726, 46489, 57007, 69849, 85211, 103871, 126119, 152987, 184955, 223349, 268898, 323384, 387830, 464587, 555168, 662619, 789084

Offset

1,6

Comments

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

Convolution of A000041 and A086971.

Links

Table of n, a(n) for n=1..53.

Index entries for related partition-counting sequences

Formula

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

Example

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.

Mathematica

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]]

Crossrefs

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

Sequence in context: A147620 A195746 A088953 * A349050 A025034 A145722

Adjacent sequences: A281609 A281610 A281611 * A281613 A281614 A281615

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 25 2017

Status

approved