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A350523 Prime numbers p such that K(p) = 0! + 1! + ... + (p-1)! == -2 (mod p). 0
2, 3, 23, 67, 227, 10331 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Table of n, a(n) for n=1..6.

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A003422, A100612.

Subsequence of A236400.

Sequence in context: A143853 A195241 A090180 * A262730 A260127 A009130

Adjacent sequences: A350520 A350521 A350522 * A350524 A350525 A350526

KEYWORD

nonn,more

AUTHOR

Luis H. Gallardo, Jan 03 2022

STATUS

approved

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Last modified January 29 10:55 EST 2023. Contains 359922 sequences. (Running on oeis4.)