A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

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

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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}]

Crossrefs

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

Sequence in context: A280127 A237977 A115339 * A036019 A018120 A240200

Adjacent sequences: A305628 A305629 A305630 * A305632 A305633 A305634

Keyword

nonn

Author

Gus Wiseman, Jun 07 2018

Status

approved