A383257 Let p = prime(n), then a(n) is the non-p-smooth part of (p-1)!+1.

1, 1, 1, 103, 329891, 2834329, 1230752346353, 336967037143579, 48869596859895986087, 10513391193507374500051862069, 8556543864909388988268015483871, 10053873697024357228864849950022572972973, 19900372762143847179161250477954046201756097561, 32674560877973951128910293168477013254334511627907

Offset

1,4

Comments

If x is an integer > 1 and p is a prime divisor of x, then a tower of x subordinate to p is an integer t such that there exists a prime divisor q of x such that q <= p, and t is the highest power of q that is a divisor of x.

If (p-1)!+1 = Product_{k} q_k^(e_k), then a(n) = Product_{k>n} q_k^(e_k). - Sean A. Irvine, May 05 2025

Let p = prime(n) and k = (p-1)!+1. If m<p, then k == 1 mod m, so a(n) = k/(p-adic valuation of k). By Wilson's theorem, a(n)<=k/p. Conjecture: a(n) = k/p^2 if n = 3, 6, or 103 and a(n) = k/p otherwise. - Chai Wah Wu, May 11 2025

Links

Table of n, a(n) for n=1..14.

Example

a(6) = 2834329 because ((13 - 1)! + 1)/w = (12! + 1)/w = (13^2*2834329)/w = 2834329, where w is the product of the towers of ((13 - 1)! + 1) subordinate to 13, w equaling 13^2.

Prog

(PARI) a(n) = my(p=prime(n), f=factor((p-1)! + 1, nextprime(p+1))); for (i=1, #f~, if (f[i, 1] <= p, f[1, 1] = 1)); factorback(f); \\ Michel Marcus, Apr 30 2025
(Python)
from sympy import prime, factorial
def A383257(n):
    p = prime(n)
    f = factorial(p-1)+1
    a, b = divmod(f, p)
    while not b:
        f = a
        a, b = divmod(f, p)
    return f # Chai Wah Wu, May 12 2025

Crossrefs

Cf. A060371, A383578.

Sequence in context: A245495 A097726 A262273 * A088584 A238490 A097014

Adjacent sequences: A383254 A383255 A383256 * A383258 A383259 A383260

Keyword

nonn

Author

Mike Jones, Apr 29 2025

Extensions

More terms from Michel Marcus, Apr 30 2025

Status

approved