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Product of numbers of elements of non-empty nonempty subsets of divisors of n.

Number of non-empty nonempty subsets of divisors of n = A100587(n).

For n = 4; divisors of 4: {1, 2, 4}; non-empty 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.

Cf. A100587, A000005, A100587, A229333.

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1, 2, 2, 24, 2, 20736, 2, 20736, 24, 20736, 2, 11501279977342425366528000000, 2, 20736, 20736, 309586821120, 2, 11501279977342425366528000000, 2, 11501279977342425366528000000, 20736, 20736, 2, 76958943408207994310451133909928754016691287539660927138035169837998325372031682525056204800000000000000000000000000000000000000000000000000000000

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

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

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allocated for Jaroslav KrizekProduct of numbers of elements of non-empty 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, 76958943408207994310451133909928754016691287539660927138035169837998325372031682525056204800000000000000000000000000000000000000000000000000000000

1,2

Number of non-empty subsets of divisors of n = A100587(n).

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

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

For n = 4; divisors of 4: {1, 2, 4}; non-empty 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.

Cf. A100587, A000005, A229333.

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Jaroslav Krizek, Sep 30 2013

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allocated for Jaroslav Krizek

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