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%I #23 Jul 20 2017 23:17:00
%S 3,13,2,1657,2,83,5,431,5,199,3
%N Primes r with A253683(n) > A253684(n) > r such that (A253683(n), A253684(n), r) is 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) must have A253683(n) > 121637. - _Felix Fröhlich_, Jun 18 2016
%C a(12) must have A253683(n) > 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(r, ", ")))))
%Y Cf. A124121, A124122.
%Y Cf. A253683, A253684.
%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
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