[seqfan] Quartets, pairs or single ?

[юрий герасимов]

Dear SeqFans, 
whether useful the following sequences in the OEIS and in what format (quartets, couples or single) to represent them there ? 
1) ltst of nonnegative noncomposite quartets (2^x - y^2, x^2 - 2^y, x, y) : 
2.......................0.....................1.....................0 
3.......................2.....................2.....................1 
7.......................7......................3.....................1 
23.....................17.....................5.....................3 
31.....................23.....................5.....................1 
103...................17.....................7......................5 
127...................47.....................7......................1 
2039.................113...................11.....................3 
8167.................137...................13.....................5 
8191..................167...................13....................1 
131063...............281...................17...................3 
524287...............359...................19...................1 
?.........................?......................?....................? 
2) list of prime quartets (2^x - y^2, x^2 - 2^y, x, y) : (23, 17, 5, 3), (103, 17, 7, 5), (2039, 113, 11, 3), (8167, 137, 13, 5), (131063, 281, 17, 3),..?
3) list of prime pairs (g, q), where g > q, g = 2^x - y^2, q = x^2 - 2^y : (3, 2), (23, 17), (31, 23), (103, 17), (127, 47), (503, 73), (2039, 113), (8167, 137), (8191, 167), (32719, 07), (131063, 281), (524287, 359), (2097143, 433), (34359738319, 1097), (562949953421231, 1889),...?
4) primes of the forh 2^x - y^2 such that x and x^2 - 2^y are both prime : 3, 7 ,23, 31, 103, 127, 503, 2039, 8167, 8191, 32719, 131063, 524287, 2097143, 34359738319, 562949953421231,..?
5) if y = 0 or y = 1 then g = 2^x - 1^2 and q = x^2 - 2^1, i. e. 
5a) primes p such that 2^p - 1 and p^2 - 2 are both prime : 2, 3, 5, 7, 13, 19, 61, 89, 107, 127, 607, 4423, 859433,...?   or
5b) primes of the form 2^p - 1 such that p and p^2 - 2 are both prime : 3, 7, 31, 127, 8191,..?   or
5c) primes of the form p^2 - 2 such that p and 2^p - 1 are both prime : 2, 7, 23, 47, 167, 359, ..?
Regards, 
Juri-Stepan Gerasimov.