A078635 Number of partitions of n into perfect powers.

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

Offset

0,5

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{k=i^j, i>=1, j>=2, excluding duplicates} 1/(1 - x^k). - Ilya Gutkovskiy, Mar 21 2017

Example

a(10)=5 since 10 can be written as 9+1, 8+1+1, 4+4+1+1, 4+1+1+1+1+1+1, or 1+1+1+1+1+1+1+1+1+1.

Mathematica

t = Union[Flatten[Table[n^k, {n, 1, 60}, {k, 2, 10}]]]; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)
With[{nn = 64}, CoefficientList[Series[Product[1/(1 - x^k), {k, Select[Range[nn], # == 1 || GCD @@ FactorInteger[#][[All, -1]] > 1 &]}], {x, 0, nn}], x]] (* Michael De Vlieger, Sep 06 2022 *)

Crossrefs

Cf. A001597.

Cf. A131799.

Sequence in context: A230447 A029078 A131799 * A286305 A046768 A363405

Adjacent sequences: A078632 A078633 A078634 * A078636 A078637 A078638

Keyword

nonn

Author

Henry Bottomley, Dec 12 2002

Status

approved