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

Author

R. J. Mathar, Jun 03 2009

Keywords

• nonn

Examples

			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.
		

Crossrefs

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

Programs

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 *)

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) ).