A179400 -id:A179400 - cp's OEIS frontend

A179678 Primes which are the fifth element of a generalized Wieferich sequence.

359, 6211, 13477, 19069, 20431, 22567, 32933, 39041

Offset: 1

M. F. Hasler, Jan 10 2011

nonn

A generalized Wieferich sequence is an increasing sequence of primes p[1],p[2],... such that a=p[n+1] is a Wieferich prime to base b=p[n], i.e., a^(b-1)=1 (mod b^2).

The present sequence is a subsequence of A179400. The numbers 728437, 1051139, 144 449603 and 5176 948723 found by M. Kamada and D. Broadhurst are also in this sequence. The last two numbers are even 7th elements of a generalized Wieferich sequence, cf. the link.

			The smallest generalized Wieferich sequence of length 5 is (3, 11, 71, 331, 359): 3^10=1 (mod 11^2), 11^70=1 (mod 71^2), 71^330=1 (mod 331^2) and 331^358=1 (mod 359^2). Therefore, a(1)=359.
It happens that after a(2)=6211, there are four subsequent elements of A179400, namely 13477, 19069, 20431 and 22567, which are also member of a sequence of length 5 and thus in this sequence.

Kevin Acres, Mike Oakes, David Broadhurst, Makoto Kamada, 1993/2011 puzzle, digest of 15 messages in primenumbers Yahoo group, Jan 8 - Jan 9, 2011.
D. Broadhurst, Re: 1993/2011 puzzle [and Puzzle 7], in primenumbers@yahoogroups.com, Jan 2011.

fp(p)={ my(a=precprime(p)); while(Mod(a, p^2)^(p-1)-1 && a=precprime(a-1), ); a }
forprime(p=1, default(primelimit), my(a=p); for(c=1, 4, (a=fp(a))||next(2)); print1(p, ", "))

A359952 Wieferich sequence where a(1) = 2.

Original entry on oeis.org

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

Robert C. Lyons, Jan 19 2023

nonn

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

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

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

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

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)

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A179678 Primes which are the fifth element of a generalized Wieferich sequence.

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