A359952 Wieferich sequence where a(1) = 2.

2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043, 5, 20771, 18043

Offset: 1

Author

Robert C. Lyons, Jan 19 2023

Keywords

• nonn

Comments

Starting with a(3), the sequence is periodic with the following cycle, which is a Wieferich triplet: 5, 20771, 18043.

Links

• Wikipedia, Wieferich sequence.
• Wikipedia, Wieferich prime.
• Wikipedia, Wieferich pair.

Crossrefs

Cf. A178871, A179400, A179678, A244550, A297846.

Programs

PARI

i=0; a=2; print1(a, ", "); while(i<100, forprime(p=2, 10^6, if(Mod(a, p^2)^(p-1)==1 && p%2!=0 && ((a-1) % p^2) && ((a+1) % p^2), print1(p, ", "); i++; a=p; break({n=1})))) \\ Michel Marcus, Jan 21 2023

Python

from sympy import nextprime
from gmpy2 import powmod
max_n = 45
a = 2
seq = [a]
for i in range(2, max_n+1):
    p = 2
    while True:
        p_squared = p*p
        if powmod(a, p-1, p_squared) == 1 and (a-1) % p_squared != 0 and (a+1) % p_squared != 0:
            seq.append(p)
            a = p
            break
        else:
            p = nextprime(p)
print(seq)