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A369375
Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).
2
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
1,2
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).
FORMULA
For n > 1, a(n) = A138290(n-1) + 1.
A208083(a(n)) = 0, for n > 0.
EXAMPLE
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.
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 *)
KEYWORD
nonn
AUTHOR
Thomas Ordowski, Jan 22 2024
EXTENSIONS
More terms from Robert G. Wilson v, Jan 22 2024
STATUS
approved