A284171 - OEIS

%I #14 Mar 22 2017 12:53:15

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

%N Number of partitions of n into distinct perfect powers (including 1).

%C Differs from the sequence A112345 which does not consider 1 as a perfect power.

%H Vaclav Kotesovec, <a href="/A284171/b284171.txt">Table of n, a(n) for n = 0..10000</a>

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

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=1} (1 + x^A001597(k)).

%F a(n) = A112345(n-1) + A112345(n).

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

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

%o (PARI) Vec((1 + x) * prod(k=1, 100, 1 + (gcd(factorint(k)[,2])>1)*x^k) + O(x^101)) \\ _Indranil Ghosh_, Mar 21 2017

%Y Cf. A001597, A078635, A112344, A112345.

%K nonn

%O 0,10

%A _Ilya Gutkovskiy_, Mar 21 2017