A056583 -id:A056583 - cp's OEIS frontend

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

Henry Bottomley, Jul 03 2000

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.

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

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.

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

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).

A056584 Solution to (n^2/a(n))^a(n) = gcd(n^n, Product_{k

Original entry on oeis.org

4, 9, 0, 25, 3, 49, 4, 3, 5, 121, 12, 169, 7, 15, 16, 289, 18, 361, 20, 21, 11, 529, 24, 25, 13, 27, 28, 841, 30, 961, 32, 33, 17, 35, 36, 1369, 19, 39, 40, 1681, 42, 1849, 44, 45, 23, 2209, 48, 49, 50, 51, 52, 2809, 54, 55, 56, 57, 29, 3481, 60, 3721, 31, 63, 64, 65

Offset: 2

Henry Bottomley, Jul 03 2000

nonn

			For n = 4, there are no integer solutions of (16/a)^a = 4, though there are two real solutions of about 14.5454938 and 0.3673156945.

Cf. A000312, A002109, A056582, A056583.

a(2) = 4, a(4) = 0, a(8) = 4, a(9) = 3; if p an odd prime: a(p) = p^2 and a(2p) = p; otherwise if n>1: a(n) = n. Apart from n = 4, a(n) = n^2/A056583(n) = log(A056582(n))/log(A056583(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Programs

Python

Formula