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Primes p in increasing order with p > A253684(n) > A253685(n) such that (p, A253684(n), A253685(n)) forms a Wieferich triple.
4

%I #27 Jul 20 2017 23:16:39

%S 71,863,1093,2281,3511,13691,20771,54787,950507,1843757,3188089

%N Primes p in increasing order with p > A253684(n) > A253685(n) such that (p, A253684(n), A253685(n)) forms a Wieferich triple.

%C In analogy to a Wieferich pair, a set of three primes p, q, r can be called a 'Wieferich triple' if its members satisfy either of the following two sets of congruences:

%C p^(q-1) == 1 (mod q^2), q^(r-1) == 1 (mod r^2), r^(p-1) == 1 (mod p^2)

%C p^(r-1) == 1 (mod r^2), r^(q-1) == 1 (mod q^2), q^(p-1) == 1 (mod p^2)

%C a(9) > 121637. - _Felix Fröhlich_, Jun 18 2016

%C a(12) > 5*10^6. - _Giovanni Resta_, Jun 20 2016

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wieferich_pair">Wieferich pair</a>

%o (PARI) forprime(p=1, , forprime(q=1, p, forprime(r=1, q, if((Mod(p, q^2)^(q-1)==1 && Mod(q, r^2)^(r-1)==1 && Mod(r, p^2)^(p-1)==1) || (Mod(p, r^2)^(r-1)==1 && Mod(r, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1), print1(p, ", ")))))

%Y Cf. A124121, A124122, A253684, A253685.

%K nonn,hard,more

%O 1,1

%A _Felix Fröhlich_, Jan 09 2015

%E a(8) from _Felix Fröhlich_, Jun 18 2016

%E Name edited by _Felix Fröhlich_, Jun 18 2016

%E a(9)-a(11) from _Giovanni Resta_, Jun 20 2016