A161148 Number of partitions of n such that each term of the partition is a squared divisor of n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 2, 6, 1, 9, 1, 8, 3, 6, 1, 16, 2, 7, 4, 12, 1, 21, 1, 15, 4, 9, 2, 39, 1, 10, 5, 25, 1, 35, 1, 24, 9, 12, 1, 76, 2, 21, 6, 32, 1, 61, 3, 38, 7, 15, 1, 174, 1, 16, 10, 46, 3, 93, 1, 50, 8, 42, 1, 231, 1, 19, 19, 60, 2, 135, 1

Offset

1,4

Links

Table of n, a(n) for n=1..79.

Formula

a(p) = 1 if p a prime (A000040).

a(2p) = A130291(n) if p=A000040(n).

a(n) = [x^n] Product_{d|n} 1/( 1-x^(d^2) ).

Example

a(n=12)=5 counts these 5 partitions of 12: 1^2+1^2+..+1^2 = 1^2+1^2+...+1^2+2^2 = 1^2+1^2+..+1^2+2^2+2^2 = 1^2+1^2+1^2+3^2=2^2+2^2+2^2. Partitions with the divisors 4, 6 or 12 do not contribute to the count because 4^2, 6^2 and 12^2 are larger than n.

Maple

a := proc(n) coeftayl(1/mul(1-x^(d^2), d=numtheory[divisors](n)), x=0, n) ; end:

Mathematica

a[n_] := SeriesCoefficient[1/Product[1-x^(d^2), {d, Divisors[n]}], {x, 0, n}];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Apr 04 2024, after Maple code *)

Crossrefs

Cf. A000040, A130291, A018818, A001156, A002635.

Sequence in context: A376719 A305974 A332267 * A143773 A323524 A354713

Adjacent sequences: A161145 A161146 A161147 * A161149 A161150 A161151

Keyword

nonn

Author

R. J. Mathar, Jun 03 2009

Status

approved