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Odd primes p such that the congruence 2^x == 1 (mod p) has no solution for 0 < x < (p - 1)/2.
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%I #11 Jul 17 2023 09:03:29

%S 3,5,7,11,13,17,19,23,29,37,41,47,53,59,61,67,71,79,83,97,101,103,107,

%T 131,137,139,149,163,167,173,179,181,191,193,197,199,211,227,239,263,

%U 269,271,293,311,313,317,347,349,359,367,373,379,383,389,401,409,419

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

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

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

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

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

%o (Python)

%o from itertools import islice

%o from sympy import nextprime, n_order

%o def A363286_gen(startvalue=3): # generator of terms >= startvalue

%o p = max(startvalue,3)-1

%o while (p:=nextprime(p)):

%o if n_order(2,p)<<1 >= p-1:

%o yield p

%o A363286_list = list(islice(A363286_gen(),30)) # _Chai Wah Wu_, Jul 17 2023

%Y Cf. A001917, A014664.

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, May 25 2023