A double Mersenne number is a number of the form M_(M_n) = 2^(2^n - 1) - 1, where M_n is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, 9223372036854775807, ... (OEIS A077585). A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne prime M_n can be prime only for prime n, a double Mersenne prime can be prime only for prime M_n, i.e., M_n a Mersenne prime. Double Mersenne numbers are prime for n = 2, 3, 5, 7, corresponding to the sequence 7, 127, 2147483647, 170141183460469231731687303715884105727, ... (OEIS A077586).
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