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Number of partitions of n into parts which are all powers of the same composite.
6

%I #6 Dec 30 2017 05:14:01

%S 0,0,0,1,1,2,2,4,5,6,6,9,9,10,11,15,15,18,18,22,23,24,24,30,31,32,34,

%T 38,38,42,42,48,49,50,51,60,60,61,62,69,69,74,74,79,82,83,83,94,95,98,

%U 99,105,105,111,112,120,121,122,122,134,134,135,138,149,150,155,155

%N Number of partitions of n into parts which are all powers of the same composite.

%C The exponents cannot all be 0.

%C Diverges from A028422 at n=20.

%H Andrew Howroyd, <a href="/A064574/b064574.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{k>=1} 1/(Product_{r>=0} 1-x^(A002808(k)^r)) - 1/(1-x). - _Andrew Howroyd_, Dec 29 2017

%e a(8)=4: 8^1, 6^1+2*6^0, 2*4^1, 4^1+4*2^0

%o (PARI) first(n)={Vec(sum(k=2, n, if(!isprime(k), 1/prod(r=0, logint(n,k), 1-x^(k^r) + O(x*x^n)) - 1/(1-x), 0)), -n)} \\ _Andrew Howroyd_, Dec 29 2017

%Y A002808, A064572, A064573, A064575, A064576, A064577, A028422.

%K easy,nonn

%O 1,6

%A _Marc LeBrun_, Sep 20 2001

%E Name clarified by _Andrew Howroyd_, Dec 29 2017