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

1, 1, 4, 1, 1728, 1, 65536, 19683, 3200000, 1, 8916100448256, 1, 13492928512, 437893890380859375, 18446744073709551616, 1, 39346408075296537575424, 1, 104857600000000000000000000

Offset: 2

Author

Henry Bottomley, Jul 03 2000

Keywords

• nonn
• easy

Comments

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.

Examples

			a(6) = gcd(46656, 86400000) = 1728.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..200
• Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.

Programs

Python

from gmpy2 import gcd
A056582_list, n = [], 1
for i in range(2,201):
    m = i**i
    A056582_list.append(int(gcd(n,m)))
    n *= m # Chai Wah Wu, Aug 21 2015

Formula

a(n) = GCD(A000312(n), A002109(n-1)).

Except for n = 4, a(n) = A056583(n)^A056584(n) = A056583(n)^(n^2/A056583(n)) = (n^2/A056584(n))^A056584(n).