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Product of divisors of n-th perfect number.
2

%I #12 Oct 17 2024 08:23:59

%S 36,21952,30019840638976,2343612817652433441167245312,

%T 68243196466656578806189194595182660833083134598062479091851170386830226748067289941548778081222656

%N Product of divisors of n-th perfect number.

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

%H Amiram Eldar, <a href="/A067584/b067584.txt">Table of n, a(n) for n = 1..8</a>

%F 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.

%F a(n) = A007955(A000396(n)). - _Amiram Eldar_, Oct 17 2024

%e 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)).

%e 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)).

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

%Y Cf. A000396, A000043, A007955.

%K nonn

%O 1,1

%A _Rick L. Shepherd_, Jan 31 2002