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A056582 Highest common factor (or GCD) of n^n and hyperfactorial(n-1), i.e., gcd(n^n, product(k^k) for k < n).

%I A056582 #18 Aug 03 2017 02:59:40
%S A056582 1,1,4,1,1728,1,65536,19683,3200000,1,8916100448256,1,13492928512,
%T A056582 437893890380859375,18446744073709551616,1,39346408075296537575424,1,
%U A056582 104857600000000000000000000
%N A056582 Highest common factor (or GCD) of n^n and hyperfactorial(n-1), i.e., gcd(n^n, product(k^k) for k < n).
%C A056582 Sequence could be defined as: a(2) = 1, a(4) = 4, a(8) = 65536, a(9) = 19683; if p an odd prime: a(p) = 1 and a(2p) = (4p)^p; otherwise if n > 1: a(n) = n^n.
%H A056582 Chai Wah Wu, <a href="/A056582/b056582.txt">Table of n, a(n) for n = 2..200</a>
%H A056582 Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537.  Zbl 1133.11012.
%F A056582 a(n) = GCD(A000312(n), A002109(n-1)).
%F A056582 Except for n = 4, a(n) = A056583(n)^A056584(n) = A056583(n)^(n^2/A056583(n)) = (n^2/A056584(n))^A056584(n).
%e A056582 a(6) = gcd(46656, 86400000) = 1728.
%o A056582 (Python)
%o A056582 from gmpy2 import gcd
%o A056582 A056582_list, n = [], 1
%o A056582 for i in range(2,201):
%o A056582     m = i**i
%o A056582     A056582_list.append(int(gcd(n,m)))
%o A056582     n *= m # _Chai Wah Wu_, Aug 21 2015
%K A056582 nonn,easy
%O A056582 2,3
%A A056582 _Henry Bottomley_, Jul 03 2000