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COMMENTS
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Primes p such that exactly one of k*2^p - 2*k + 1 and k*2^p + 2*k - 1 is a prime:
k = 2: this sequence;
k = 3: 5, 13, 19, 29, 31, 109, 139, 271, 379, 1553, ...
k = 4: 2, 37, ...
k = 5: 3, 5, 7, 17, 19, 23, 41, 61, 67, 151, 157, 313, 4111, 6337, ...
k = 6: 2, 5, 7, 11, 19, 29, 149, 191, 373, 449, 983, 1667, 1973, ...
k = 7: 2, 3, 5, 7, 11, 13, 29, 43, 61, 97, 109, 127, 131, 239, 461, 1153, ...
k = 8: 3, 11, 19, 23, 29, 37, 43, 97, 193, 307, 617, 1847, ...
k = 9: 3, 5, 23, 41, 61, 71, 97, 131, 157, 863, 3119, ...
k = 10: 2, 3, 13, ...
...
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