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%I #18 Sep 06 2022 13:32:05
%S 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,
%T 43,48,50,52,62,70,75,79,92,100,105,109,126,140,148,157,177,194,202,
%U 211,237,261,276,290,324,351,370,384,424,462,489,514,562,609,640,670,728
%N Number of partitions of n into perfect powers.
%H Michael De Vlieger, <a href="/A078635/b078635.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: Product_{k=i^j, i>=1, j>=2, excluding duplicates} 1/(1 - x^k). - _Ilya Gutkovskiy_, Mar 21 2017
%e 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.
%t 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*)
%t a[n_] := Length@p@n; a /@ Range[0, 80]
%t (* _Clark Kimberling_, Mar 09 2014 *)
%t 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 *)
%Y Cf. A001597.
%Y Cf. A131799.
%K nonn
%O 0,5
%A _Henry Bottomley_, Dec 12 2002
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