A064572 - OEIS

%I #36 Dec 30 2017 05:13:17

%S 0,1,2,5,6,10,11,17,20,26,27,38,39,47,51,65,66,82,83,102,107,123,124,

%T 153,156,178,185,216,217,254,255,297,304,342,346,408,409,457,466,535,

%U 536,609,610,690,704,780,781,895,898,998,1009,1130,1131,1263,1268,1418

%N Number of ways to partition n into parts which are all powers of some integer k.

%C Number of ways to partition n as Sum_i k^e_i, where the exponents e_i are not all 0.

%C The exponents cannot all be 0, e.g. a(2)=1 arises from 2^1, and does not include 2^0+2^0. - _Shujing Lyu_, Apr 23 2016

%H Andrew Howroyd, <a href="/A064572/b064572.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..220 from Shujing Lyu)

%F G.f.: Sum_{k>=2} 1/(Product_{r>=0} 1-x^(k^r)) - 1/(1-x). - _Andrew Howroyd_, Dec 29 2017

%e a(4)=5: 4^1, 3^1+3^0, 2^2, 2*2^1, 2^1+2*2^0.

%o (PARI) first(n)={Vec(sum(k=2, n, 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 Cf. A028422, A064573, A064574, A064575, A064576, A064577, A292477.

%K easy,nonn

%O 1,3

%A _Marc LeBrun_, Sep 20 2001