A054685 Number of partitions of n into distinct prime powers (1 not considered a power).

1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 5, 6, 7, 7, 10, 9, 12, 12, 15, 15, 18, 19, 22, 24, 26, 30, 32, 36, 39, 43, 48, 51, 57, 61, 68, 73, 79, 87, 93, 103, 108, 121, 127, 140, 148, 162, 173, 187, 200, 215, 232, 247, 266, 283, 306, 324, 348, 371, 397, 423, 450, 480, 512, 543, 579, 614

Offset: 0

Author

David W. Wilson, Apr 19 2000

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)

Crossrefs

Cf. A051613.

Cf. A106244.

Cf. A000961.

Programs

Haskell

import Data.MemoCombinators (memo2, integral)
a054685 n = a054685_list !! n
a054685_list = map (p' 2) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < pp then 0 else p' (k + 1) (m - pp) + p' (k + 1) m
           where pp = a000961 k
-- Reinhard Zumkeller, Nov 23 2015

Mathematica

CoefficientList[Series[Product[Product[1 +x^(Prime[n]^k), {k, 1, 9}], {n, 1, 25}], {x, 0, 100}], x] (* G. C. Greubel, May 09 2019 *)

Formula

G.f.: Product_{p prime} Product_{k >= 1} (1 + x^(p^k)).