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%I #26 Nov 02 2023 12:30:04
%S 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,
%T 32,36,39,43,48,51,57,61,68,73,79,87,93,103,108,121,127,140,148,162,
%U 173,187,200,215,232,247,266,283,306,324,348,371,397,423,450,480,512,543,579,614
%N Number of partitions of n into distinct prime powers (1 not considered a power).
%H Reinhard Zumkeller, <a href="/A054685/b054685.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)
%F G.f.: Product_{p prime} Product_{k >= 1} (1 + x^(p^k)).
%t 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 *)
%o (Haskell)
%o import Data.MemoCombinators (memo2, integral)
%o a054685 n = a054685_list !! n
%o a054685_list = map (p' 2) [0..] where
%o p' = memo2 integral integral p
%o p _ 0 = 1
%o p k m = if m < pp then 0 else p' (k + 1) (m - pp) + p' (k + 1) m
%o where pp = a000961 k
%o -- _Reinhard Zumkeller_, Nov 23 2015
%Y Cf. A051613.
%Y Cf. A106244.
%Y Cf. A000961.
%K nonn
%O 0,6
%A _David W. Wilson_, Apr 19 2000
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