Mersenne exponents that are part of a twin prime pair.
3, 5, 7, 13, 17, 19, 31, 61, 107, 521, 1279, 4423, 110503, 132049, 20996011, 24036583
1
Boklan and Conway conjecture that this sequence is finite. See S000870 for the complement.
T. D. Noe, Plot of 16 terms
Kent D. Boklan and John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv 1605.01371 (May 04 2016)
(Mma) p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657}; Select[p, PrimeQ[# - 2] || PrimeQ[# + 2] &]
Cf. A000043 (Mersenne exponents), S000870.
nonn,more
T. D. Noe, May 04 2016