cp's OEIS Frontend

A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

%I A305630 #9 Jun 09 2018 01:51:15
%S A305630 1,1,2,3,4,6,9,12,16,21,28,36,48,61,78,99,124,156,195,241,299,367,450,
%T A305630 549,670,811,982,1183,1422,1704,2040,2431,2894,3435,4070,4811,5679,
%U A305630 6684,7858,9217,10797,12623,14738,17174,19988,23225,26951,31227,36141,41759
%N A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).
%C A305630 a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).
%H A305630 Alois P. Heinz, <a href="/A305630/b305630.txt">Table of n, a(n) for n = 0..10000</a>
%e A305630 The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).
%p A305630 q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
%p A305630 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A305630      `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A305630     end:
%p A305630 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jun 07 2018
%t A305630 nn=20;
%t A305630 radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
%t A305630 ser=Product[1/(1-x^p),{p,Select[Range[nn],radQ]}];
%t A305630 Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]
%Y A305630 Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305631-A305635.
%K A305630 nonn
%O A305630 0,3
%A A305630 _Gus Wiseman_, Jun 07 2018