A350523 Prime numbers p such that K(p) = 0! + 1! + ... + (p-1)! == -2 (mod p).

2, 3, 23, 67, 227, 10331

Offset: 1

Author

Luis H. Gallardo, Jan 03 2022

Keywords

• nonn
• more

Comments

The Kurepa Conjecture says that K(p) is nonzero in F_p, the finite field with p elements. Primes for which K(p) takes some fixed nonzero value in F_p might have some interest.

No further terms < 6*10^6. - Michael S. Branicky, Jan 03 2022

Links

• Vladica Andrejić, Alin Bostan and Milos Tatarevic, Improved algorithms for left factorial residues, Information Processing Letters, Vol. 167 (2021), Article ID 106078, 4 p.; arXiv preprint, arXiv:1904.09196 [math.NT], 2019-2020.

Crossrefs

Cf. A003422, A100612.

Subsequence of A236400.

Programs

Mathematica

q[p_] := PrimeQ[p] && Divisible[Sum[k!, {k, 0, p - 1}] + 2, p]; Select[Range[230], q] (* Amiram Eldar, Jan 03 2022 *)

Python

from sympy import isprime
def K(n):
    ans, f = 0, 1
    for i in range(1, n+1):
        ans += f%n
        f = (f*i)%n
    return ans%n
def ok(n): return isprime(n) and (K(n) + 2)%n == 0
print([k for k in range(11000) if ok(k)]) # Michael S. Branicky, Jan 03 2022

Python

# faster version for initial segment of sequence
from sympy import isprime
def afind(limit):
    f = 1 # (p-1)!
    s = 2 # sum(0! + 1! + ... + (p-1)!)
    for p in range(2, limit+1):
        if isprime(p) and s%p == p-2:
            print(p, end=", ")
        s += f*p
        f *= p
afind(11000) # Michael S. Branicky, Jan 03 2022