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A305631
Expansion of Product_{r not a perfect power} 1/(1 - x^r).
11
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, 171, 201, 239, 282, 336, 391, 463, 541, 635, 741, 868, 1005, 1174, 1359, 1580, 1826, 2115, 2436, 2814, 3237, 3726, 4276, 4914, 5618, 6445, 7359, 8414, 9594, 10947, 12453
OFFSET
0,6
COMMENTS
a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).
LINKS
EXAMPLE
The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).
MAPLE
q:= n-> is(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=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All, 2]]==1;
ser=Product[1/(1-x^p), {p, Select[Range[nn], wadQ]}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, 0, nn}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 07 2018
STATUS
approved