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

0,5

Links

Table of n, a(n) for n=0..75.

Eric Weisstein's World of Mathematics, Powerful Number

Index entries for sequences related to powerful numbers

Index entries for sequences related to partitions

Formula

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

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

Example

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

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]

Crossrefs

Cf. A001694, A078635, A286320.

Sequence in context: A029078 A131799 A078635 * A046768 A363405 A274151

Adjacent sequences: A286302 A286303 A286304 * A286306 A286307 A286308

Keyword

nonn

Author

Ilya Gutkovskiy, May 12 2017

Status

approved