A067584 Product of divisors of n-th perfect number.

36, 21952, 30019840638976, 2343612817652433441167245312, 68243196466656578806189194595182660833083134598062479091851170386830226748067289941548778081222656

Offset: 1

Author

Rick L. Shepherd, Jan 31 2002

Keywords

• nonn

Comments

This is a subsequence of A007955, product of divisors of n.

Examples

			a(2) = 21952 because a(2) = 1*2*4*7*14*28=21952 (= 28^3) is the product of all (3*2) divisors of the 2nd perfect number, 28 (= 2^(3-1) * (2^3 - 1)).
a(6), a 169 decimal digit number (= 8589869056^17), is the product of all (17*2) divisors of the 6th perfect number, 8589869056 (= 2^(17-1) * (2^17 - 1)).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..8

Crossrefs

Cf. A000396, A000043, A007955.

Programs

Mathematica

f[n_] := n^(DivisorSigma[0, n]/2); f /@ PerfectNumber[Range[5]] (* Amiram Eldar, Oct 17 2024 *)

Formula

a(n) = A000396(n)^A000043(n) whenever A000396(n) is an even perfect number. (Proof of the existence of odd perfect numbers is a well-known open problem.) A000043(n) is a Mersenne exponent.

a(n) = A007955(A000396(n)). - Amiram Eldar, Oct 17 2024