A263686 Smallest prime factor of double Mersenne numbers.

7, 127, 2147483647, 170141183460469231731687303715884105727, 338193759479, 231733529, 62914441, 295257526626031

Offset: 1

Author

Arkadiusz Wesolowski, Oct 23 2015

Keywords

• nonn
• hard
• more

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

Crossrefs

Cf. A000043, A000668, A001348, A020639, A049479, A077586, A122094. Subsequence of A016047. Subsequence of A309130.

Programs

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

Formula

a(n) = spf(MM(A000043(n))) = A049479(A000668(n)), where spf = A020639 is the smallest prime factor, A049479 = spf o M, M(p) = 2^p-1 = A000225(p), MM = M o M = A077585, A000668(n) = M(A000043(n)), A000043 are the Mersenne prime exponents. - M. F. Hasler, Mar 01 2025