login
Number of partitions of n into distinct powerful parts (A001694).
1

%I #29 May 14 2017 00:01:33

%S 1,1,0,0,1,1,0,0,1,2,1,0,1,2,1,0,1,2,1,0,1,2,1,0,1,3,2,1,2,3,2,1,2,3,

%T 3,2,4,5,3,2,4,5,3,2,4,6,4,2,4,7,5,2,5,8,5,2,5,8,6,3,5,10,8,4,6,10,8,

%U 4,6,10,9,5,8,12,10,6,9,13,10,6,9,15,12,7,10,17,14,7,11,18,15,8,11,18,16,9,11,20,18,10,13

%N Number of partitions of n into distinct powerful parts (A001694).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: Product_{k>=1} (1 + x^A001694(k)).

%F a(n) = A284171(n) for n < 72.

%e a(25) = 3 because we have [25], [16, 9] and [16, 8, 1].

%t nmax = 100; CoefficientList[Series[(1 + x) Product[1 + Boole[Min@ FactorInteger[k][[All, 2]] > 1] x^k, {k, 2, nmax}], {x, 0, nmax}], x]

%Y Cf. A001694, A284171, A286305.

%K nonn

%O 0,10

%A _Ilya Gutkovskiy_, May 12 2017