A305631 - OEIS

%I #9 Jun 09 2018 01:07:56

%S 1,0,1,1,1,2,3,3,4,5,7,8,12,13,17,21,25,32,39,46,58,68,83,99,121,141,

%T 171,201,239,282,336,391,463,541,635,741,868,1005,1174,1359,1580,1826,

%U 2115,2436,2814,3237,3726,4276,4914,5618,6445,7359,8414,9594,10947,12453

%N Expansion of Product_{r not a perfect power} 1/(1 - x^r).

%C a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).

%H Alois P. Heinz, <a href="/A305631/b305631.txt">Table of n, a(n) for n = 0..10000</a>

%e The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).

%p q:= n-> is(1=igcd(map(i-> i[2], ifactors(n)[2])[])):

%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(

%p      `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Jun 07 2018

%t nn=100;

%t wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;

%t ser=Product[1/(1-x^p),{p,Select[Range[nn],wadQ]}];

%t Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

%Y Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

%K nonn

%O 0,6

%A _Gus Wiseman_, Jun 07 2018