A051696 Greatest common divisor of n! and n^n.

1, 2, 3, 8, 5, 144, 7, 128, 81, 6400, 11, 248832, 13, 100352, 91125, 32768, 17, 429981696, 19, 163840000, 6751269, 63438848, 23, 247669456896, 15625, 1417674752, 1594323, 80564191232, 29, 25076532510720000000, 31, 2147483648, 19098395217, 1241245548544, 6565234375

Offset

1,2

Comments

a(n) also equals the smallest positive integer such that lcm(a(1), a(2), a(3), ... a(n)) = n!, for every positive integer n. - Leroy Quet, Apr 28 2007

Links

T. D. Noe, Table of n, a(n) for n = 1..500

Formula

a(n) = Product_{p|n} p^(sum{k >= 1} floor(n/p^k)), where the product runs over the distinct primes p that divide n. - Leroy Quet, Apr 28 2007

a(n) = n*A062763(n). - R. J. Mathar, Mar 11 2017

a(n) = (numerator of B(n, 1/n))/n^(n - 1), where B(.,.) is the Euler beta function. - Arkadiusz Wesolowski, Nov 22 2017

a(p) = p for p prime. - Peter Luschny, Nov 29 2017

Example

a(4) = 8 since 4! = 24 and 4^4 = 256 and gcd(24, 256) = 8.
lcm(a(1), a(2), a(3), a(4), a(5), a(6)) = lcm(1, 2, 3, 8, 5, 144) = 6! = 720. (See comment.)

Maple

seq(igcd(n!, n^n), n=1..35); # Peter Luschny, Nov 29 2017

Mathematica

Table[GCD[n!, n^n], {n, 40}] (* Harvey P. Dale, Oct 20 2011 *)
(* Alternative: *)
Table[Numerator@Beta[n, 1/n]/n^(n - 1), {n, 35}] (* Arkadiusz Wesolowski, Nov 22 2017 *)

Prog

(Python)
from sympy import factorint
from sympy.ntheory.factor_ import digits
def A051696(n):
    c = 1
    for p, e in factorint(n).items():
        c *= p**min(e*n, (n-sum(digits(n, p)[1:]))//(p-1))
    return c # Chai Wah Wu, May 02 2026
(PARI) a(n) = gcd(n!, n^n); \\ Bruce Nye, Jun 23 2026

Crossrefs

Sequence in context: A170911 A067911 A243103 * A066570 A073656 A047930

Adjacent sequences: A051693 A051694 A051695 * A051697 A051698 A051699

Keyword

nonn,easy,nice,changed

Author

Leroy Quet

Extensions

More terms from James Sellers, Dec 08 1999

Status

approved