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A305630
Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).
11
1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 48, 61, 78, 99, 124, 156, 195, 241, 299, 367, 450, 549, 670, 811, 982, 1183, 1422, 1704, 2040, 2431, 2894, 3435, 4070, 4811, 5679, 6684, 7858, 9217, 10797, 12623, 14738, 17174, 19988, 23225, 26951, 31227, 36141, 41759
OFFSET
0,3
COMMENTS
a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).
LINKS
EXAMPLE
The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).
MAPLE
q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
`if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
seq(a(n), n=0..60); # Alois P. Heinz, Jun 07 2018
MATHEMATICA
nn=20;
radQ[n_]:=Or[n==1, GCD@@FactorInteger[n][[All, 2]]==1];
ser=Product[1/(1-x^p), {p, Select[Range[nn], radQ]}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, 0, nn}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 07 2018
STATUS
approved