A383578 Let p = prime(n), then a(n) is the p-smooth part of (p-1)! + 1.

2, 3, 25, 7, 11, 169, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset

1,1

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

a(n) > p if and only if p is a Wilson prime (see A007540). - Robert Israel, Mar 12 2026

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = ((prime(n) - 1)! + 1) / A383257(n).

Example

a(6) = 169 because the prime factorization of ((13 - 1)! + 1) is 13^2*2834329, and 13^2 = 169.

Maple

f:= proc(n) local p;
   p:= ithprime(n);
   p^padic:-ordp((p-1)!+1, p)
end proc:
map(f, [$1..100]); # Robert Israel, Mar 12 2026

Prog

(PARI) a(n) = my(p=prime(n), x=(p-1)! + 1, f=factor((p-1)! + 1, nextprime(p+1))); for (i=1, #f~, if (f[i, 1] <= p, f[1, 1] = 1)); x/factorback(f); \\ Michel Marcus, Apr 30 2025
(Python)
from sympy import prime, factorial
def A383578(n):
    p, c = prime(n), 1
    f = factorial(p-1)+1
    a, b = divmod(f, p)
    while not b:
        c *= p
        f = a
        a, b = divmod(f, p)
    return c # Chai Wah Wu, May 12 2025

Crossrefs

Cf. A007540, A060371, A383257.

Sequence in context: A291262 A307922 A390836 * A048674 A320223 A296275

Adjacent sequences: A383575 A383576 A383577 * A383579 A383580 A383581

Keyword

nonn

Author

Mike Jones, Apr 30 2025

Extensions

More terms from Michel Marcus, Apr 30 2025

Status

approved