A286305 Number of partitions of n into powerful parts (A001694).

1, 1, 1, 1, 2, 2, 2, 2, 4, 5, 5, 5, 7, 8, 8, 8, 12, 14, 15, 15, 19, 21, 22, 22, 28, 33, 35, 37, 43, 48, 50, 52, 62, 70, 75, 79, 92, 100, 105, 109, 126, 140, 148, 157, 177, 194, 202, 211, 237, 261, 276, 290, 324, 351, 370, 384, 424, 462, 489, 514, 562, 609, 640, 670, 728, 787, 831, 873, 948, 1016, 1071, 1118, 1210, 1296, 1366, 1433

Offset: 0

Author

Ilya Gutkovskiy, May 12 2017

Keywords

• nonn

Examples

			a(8) = 4 because we have [8], [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].
		

Links

• Eric Weisstein's World of Mathematics, Powerful Number
• Index entries for sequences related to powerful numbers
• Index entries for sequences related to partitions

Crossrefs

Cf. A001694, A078635, A286320.

Programs

Mathematica

nmax = 75; CoefficientList[Series[1/(1 - x) Product[1/(1 - Boole[Min@ FactorInteger[k][[All, 2]] > 1] x^k), {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^A001694(k)).

a(n) = A078635(n) for n < 72.