A281611 Expansion of Sum_{p prime, i>=2} x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).

0, 0, 0, 1, 1, 2, 3, 7, 10, 16, 23, 36, 50, 73, 100, 144, 193, 267, 355, 481, 631, 838, 1088, 1426, 1833, 2368, 3019, 3861, 4879, 6178, 7751, 9737, 12131, 15120, 18721, 23181, 28535, 35110, 42991, 52606, 64090, 78015, 94609, 114621, 138398, 166927, 200737, 241131, 288864, 345649, 412592, 491931

Offset: 1

Author

Ilya Gutkovskiy, Jan 25 2017

Keywords

• nonn

Comments

Total number of proper prime power parts (A246547) in all partitions of n.

Examples

			a(6) = 2 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 0 + 0 + 1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 2.
		

Links

• Index entries for related partition-counting sequences

Crossrefs

Cf. A000041, A037032, A073335, A246547.

Programs

Mathematica

nmax = 52; Rest[CoefficientList[Series[Sum[Sign[PrimeOmega[i] - 1] Floor[1/PrimeNu[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{p prime, i>=2} x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).

a(n) = A073335(n) - A037032(n).