A001917 -id:A001917 - cp's OEIS frontend

A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

364, 1755

Offset: 1

Felix Fröhlich, May 30 2018

Meissner discovered the congruence 2^364 == 1 (mod 1093^2) and thus proved that 1093 is a Wieferich prime, i.e., a term of A001220 (cf. Meissner, 1913).
Later, Beeger discovered the congruence 2^1755 == 1 (mod 3511^2) and proved that 3511 is also a Wieferich prime (cf. Beeger, 1922).
Let b(n) = (A001220(n)-1)/a(n). Then b(1) = 3 and b(2) = 2.
From the fact that a(1) and a(2) are composite it follows that A001220(1) = 1093 and A001220(2) = 3511 do not divide any terms of A001348 (cf. Dobson).
Curiously, both 364 and 1755 are repdigits in some base. 364 = 444 in base 9 and 1755 = 3333 in base 8. Compare this with Dobson's observation that 1092 and 3510 are 444 in base 16 and 6666 in base 8, respectively (cf. Dobson).

N. G. W. H. Beeger, On a new case of the congruence 2^p-1 == 1 (mod p^2), Messenger of Mathematics 51 (1922), 149-150.
J. B. Dobson, A note on the two known Wieferich primes
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

Cf. A001220, A001348, A014664, A243905, A282552, A282902.

forprime(p=1, , if(Mod(2, p^2)^(p-1)==1, print1(znorder(Mod(2, p^2)), ", ")))

a(n) = A014664(A000720(A001220(n))) = A243905(A000720(A001220(n))). [Corrected by Jianing Song, Sep 20 2019]

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

Original entry on oeis.org

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

Arkadiusz Wesolowski, May 25 2023

nonn

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

Cf. A001917, A014664.

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

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

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

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

cp's OEIS Frontend

A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

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