A281906 Expansion of Sum_{p prime, i>=1} p^i*x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).

0, 2, 5, 13, 23, 41, 69, 119, 185, 283, 425, 625, 903, 1285, 1799, 2517, 3450, 4699, 6340, 8490, 11264, 14870, 19485, 25390, 32897, 42395, 54372, 69408, 88210, 111612, 140717, 176738, 221135, 275776, 342790, 424743, 524765, 646420, 794109, 972967, 1189105, 1449577, 1763097, 2139394, 2590349, 3129633, 3773546, 4540645

Offset: 1

Author

Ilya Gutkovskiy, Feb 01 2017

Keywords

• nonn

Comments

Total sum of prime power parts (1 excluded) in all partitions of n.

Convolution of the sequences A000041 and A023889.

Examples

			a(5) = 23 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 5 + 4 + 3 + 2 + 3 + 2 + 2 + 2 = 23.
		

Links

• Index entries for related partition-counting sequences

Crossrefs

Cf. A000041, A023889, A066186, A073118, A246655.

Programs

Mathematica

nmax = 48; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[i]] 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>=1} p^i*x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).