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%I #25 Jul 20 2017 23:15:59
%S 1093,1006003,1645333507,4871,318917,18787
%N Least prime p such that (p,q) is a Double Wieferich prime pair for q=A124121(n).
%C Double Wieferich prime pairs are pairs of primes (p, q) such that q^(p-1) == 1 (mod p^2) and p^(q-1) == 1 (mod q^2). This sequence gives the (least) value of p corresponding to the q's listed in increasing order (and without multiplicity) in A124121.
%C This is just the list of known pairs: there may be gaps.
%C Currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). - _Alexander Adamchuk_, Mar 10 2007
%H Y. F. Bilu, <a href="http://archive.numdam.org/ARCHIVE/SB/SB_2002-2003__45_/SB_2002-2003__45__1_0/SB_2002-2003__45__1_0.pdf">Catalan's Conjecture</a>, Séminaire Bourbaki, 45 (2002-2003), pp. 1-26.
%H Michael Mossinghoff, <a href="http://www.cecm.sfu.ca/~mjm/WieferichBarker">Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices</a>, as of Feb 12 2009.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wieferich_pair">Wieferich pair</a>
%o (PARI) /* The following (highly unoptimized) code misses the value a(3) but prints all other values in less than 30 seconds. */
%o default(primelimit, 1010000); forprime(q=1, default(primelimit), forprime(p=q+1, default(primelimit), Mod(p, q^2)^(q-1)-1 & next; Mod(q, p^2)^(p-1)-1 | print1( p", ") | break)) \\ _M. F. Hasler_, Oct 08 2011
%o (PARI) a(n) = my(q=prime(n), p=2); while(Mod(p, q^2)^(q-1)!=1 || Mod(q, p^2)^(p-1)!=1, p=nextprime(p+1)); p \\ _Felix Fröhlich_, Jan 04 2016
%Y See A124121 for values of q.
%Y Cf. A196511, A196733.
%K nonn,hard,more
%O 1,1
%A _N. J. A. Sloane_, following an email from _Robert G. Wilson v_, Nov 30 2006
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