A229334 Product of numbers of elements of nonempty subsets of divisors of n.

1, 2, 2, 24, 2, 20736, 2, 20736, 24, 20736, 2, 11501279977342425366528000000, 2, 20736, 20736, 309586821120, 2, 11501279977342425366528000000, 2, 11501279977342425366528000000, 20736, 20736, 2

Offset: 1

Author

Jaroslav Krizek, Sep 30 2013

Keywords

• nonn

Comments

Number of nonempty subsets of divisors of n = A100587(n).

Also product of sizes of all the subsets of set of divisors of n.

Examples

			For n = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; product of numbers of elements of subsets = 1*1*1*2*2*2*3 = 24.
For n = 4; tau(4) = 3; a(4) = [1^(3!/((3-1)!*1!))] * [2^(3!/((3-2)!*2!))] * [3^(3!/((3-3)!*3!))] = 1^3 * 2^3 * 3^1 = 24.
		

Crossrefs

Cf. A000005, A100587, A229333.

Programs

Mathematica

Table[Times @@ Rest[Length /@ Subsets[Divisors[n]]], {n, 23}] (* T. D. Noe, Oct 01 2013 *)

Formula

a(n) = product[k=1..tau(n)] k^C(tau(n),k) = product[k=1..tau(n)] k^(tau(n)!/((tau(n)-k)!*k!)).