A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

1, 7, 15, 23, 27, 31, 37, 39, 43, 55, 58, 63, 71, 79, 82, 91, 95, 111, 123, 127, 133, 135, 139, 143, 148, 151, 159, 167, 169, 172, 173, 175, 179, 183, 191, 195, 199, 207, 211, 223, 239, 255, 286, 295, 313, 316, 319, 335, 337, 351, 367, 373, 383, 406, 415, 417, 433, 435, 447, 455, 461, 463, 479

Offset: 1

Author

Thomas Ordowski, Jan 22 2024

Keywords

• nonn

Comments

Integers m > 0 such that 2^m-1 - 2^n is not prime for every natural n < m.

For m > 2, a number m is a term of this sequence if and only if A208083(m) = 0.

All Mersenne number m = 2^k-1 for k > 2 are in this sequence. The proof is below.

Cf. A138290 (see Chai Wah Wu's conjecture in the third comment). By Crocker's (1971) theorem: if m > 2 and a <> b, then 2^(2^m)-1 - 2^a - 2^b is not prime.

If a = 2^m-1, then b < a, so for m > 2, 2^(2^m-1)-1 is a de Polignac number, QED.

Note that 2^(2^m-1)-1 - 2^n is divisible by some prime factor of 2^(2^m)-1.

Prime numbers of this sequence are Mersenne primes > 3, and many other primes.

Conjecture: if n > 5, then |2^(2^n-1)-1 - 2^m| is not prime for every m > 0.

If so, then by the dual Riesel conjecture, 2^(2^n-1)-1 is a (dual) Riesel number, i.e., if n > 5, then (2^(2^n-1)-1)2^m-1 is composite for every integer m > 0.

For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number.

It seems that the natural density of these numbers is about twice as high as the density of de Polignac numbers.

For many terms m, 2m+1 is also in this sequence. By iteration (x -> 2x+1), the subsequence b(n) = (m+1)2^n-1, for n >= 0, is infinite if m = 7 (which has already been proven) and probably if m = 27 (which is hard to prove).

Examples

			7 is a term since {2^7-1-2, 2^7-1-2^2, 2^7-1-2^3, 2^7-1-2^4, 2^7-1-2^5, 2^7-1-2^6} = {125, 123, 119, 111, 95, 63} and all six of these numbers are composite.
Note that both 2^148-1 and 2^148+1 are de Polignac numbers.
		

Crossrefs

Cf. A000225, A006285, A000668, A101036, A138290, A208083, A232565, A369378.

Programs

Mathematica

fQ[n_] := Block[{k = n -1}, While[k > 1 && !PrimeQ[2^n -1 -2^k], k--]; k == 1]; Select[ Range[3, 450], fQ] (* Robert G. Wilson v, Jan 22 2024 *)

Formula

For n > 1, a(n) = A138290(n-1) + 1.

A208083(a(n)) = 0, for n > 0.

Extensions

More terms from Robert G. Wilson v, Jan 22 2024