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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 primes q which are the lesser member (listed second) of such pairs, in increasing order, but without multiplicity:

For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). In this sequence, 5 is only listed once, as a(3), and only the lesser value p=1645333507 is listed as A124122(3).

This is just the list of known pairs: there may be gaps, i.e. missing primes.

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.

This is just the list of known pairs: there may be gaps.

Currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). - Alexander Adamchuk, Mar 10 2007

Computed mainly by M. Oakes and D. Broadhurst, cf. link.

For all the listed terms, except for a(4)=2452757 and a(11)=22796069, p=(q-1)/2, which implies that they are "safe primes", cf. A005385. For a(4) and a(11), p=(q-1)/4.

The subsequence of terms of the form q=2p+1 is A196733. As of today it is complete up to its 51st term, ~2e8. - M. F. Hasler, Oct 05 2011