A152217 Primes p == 1 (mod 3) such that ((p-1)/3)! == 1 (mod p).

3571, 4219, 13669, 25117, 55897, 89269, 102121, 170647, 231019, 246247, 251431, 347821, 360187, 398581, 452797, 603457, 611557, 879667, 909151, 1021417, 1074607, 1253887, 1653919, 1667311, 2079169, 2310019, 2421907, 2603077, 2619871, 2843107, 2949217, 4033321, 4117237, 4202017

Offset

1,1

Comments

The Wilson theorem states that p is prime if and only if (p-1)! == -1 (mod p). If p == 3 (mod 4) then ((p-1)/2)! == +- 1 (mod p).

References

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.2.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..686

J. B. Cosgrave, Jacobi.

Wikipedia, Wilson's theorem.

Example

For n = 1 the prime a(1) = 3571 divides 1190! - 1.

Mathematica

p=7; lst = {}; While[p<17500000, If[ PrimeQ[p] && Mod[((p-1)/3)!, p] == 1, AppendTo[lst, p]; Print[p]]; p+=6] (* Robert G. Wilson v, Dec 19 2025 *)

Prog

(PARI) forprime(p=2, 30000, if(p%3==1 & ((p-1)/3)!%p==1, print1(p, ", ")))
(Python)
from sympy import isprime
from itertools import count
fact, m = 1, 2
for k in count(7, 6):
    fact *= m*(m-1)
    m += 2
    if isprime(k) and fact%k == 1:
        print(k, end=", ")
# Michael S. Branicky, Dec 21 2025

Crossrefs

Seems to be a subsequence of A002407 and therefore of A003215 (differences of consecutive cubes). See also A058302 and A055939 for the sequences corresponding to ((p-1)/2)! == +- 1 (mod p).

Sequence in context: A204417 A204410 A204409 * A004932 A004952 A004972

Adjacent sequences: A152214 A152215 A152216 * A152218 A152219 A152220

Keyword

nonn

Author

Francois Brunault (brunault(AT)gmail.com), Nov 29 2008

Extensions

a(12) onward from Robert G. Wilson v, Dec 19 2025

Status

approved