OFFSET
1,1
COMMENTS
A double Mersenne number is a Mersenne number of the form 2^(2^p - 1) - 1, where p is a Mersenne exponent (A000043).
From M. F. Hasler, Feb 28 2025: (Start)
The prime factors of Mersenne numbers 2^q - 1 must be of the form 2*q*k + 1.
The four smallest double Mersenne numbers (p = 2, 3, 5, 7 => q = 3, 7, 31, 127) are prime, so their smallest prime factor is equal to themselves, a(n) = M(q). This is equivalent to k = (2^(q-1)-1)/q, which is almost as large as M(q) itself: k = 1, 9 and 34636833 for the first three terms, and for q = 127, k has just three digits less than M(q) = a(4) itself. The prime p = 11 is not a Mersenne exponent.
The fifth term, a(5) = 2*(2^13-1)*k + 1 with k = 20644229 (which is prime) is the first proper divisor of the respective M(q), as are the next three, corresponding to p = 17, 19 and 31.
For p = 61, M(q) has 694127911065419642 digits, and so far no factor is known, but it is known that it has no factor less than 10^36. (End)
LINKS
Double Mersennes Prime Search, History
Wikipedia, Double Mersenne number
FORMULA
PROG
(PARI) forprime(p=2, , q=2^p-1; !ispseudoprime(q) && next(); if(ispseudoprime(2^q-1), print1(2^q-1, ", "); next()); forstep(r=2*q+1, +oo, 2*q, !ispseudoprime(r) && next(); if(Mod(2, r)^q-1 == 0, print1(r, ", "); next(2)))) \\ Jeppe Stig Nielsen, Aug 28 2019
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Arkadiusz Wesolowski, Oct 23 2015
STATUS
approved