A363286 Odd primes p such that the congruence 2^x == 1 (mod p) has no solution for 0 < x < (p - 1)/2.

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 97, 101, 103, 107, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 313, 317, 347, 349, 359, 367, 373, 379, 383, 389, 401, 409, 419

Offset: 1

Author

Arkadiusz Wesolowski, May 25 2023

Keywords

• nonn

Comments

An odd prime p belongs to this sequence if and only if A001917(A000720(p)) is equal to 1 or 2.

Crossrefs

Cf. A001917, A014664.

Programs

Magma

[p: p in [3..419 by 2] | IsPrime(p) and (p-1)/Modorder(2, p) le 2];

PARI

isok(p) = p%2 && isprime(p) && (p-1)/znorder(Mod(2, p))<=2;

Python

from itertools import islice
from sympy import nextprime, n_order
def A363286_gen(startvalue=3): # generator of terms >= startvalue
    p = max(startvalue,3)-1
    while (p:=nextprime(p)):
        if n_order(2,p)<<1 >= p-1:
            yield p
A363286_list = list(islice(A363286_gen(),30)) # Chai Wah Wu, Jul 17 2023

Formula

a(n) ~ (3/2)*n*log((3/2)*n).