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

%I A369375 #61 Mar 19 2024 12:41:02
%S A369375 1,7,15,23,27,31,37,39,43,55,58,63,71,79,82,91,95,111,123,127,133,135,
%T A369375 139,143,148,151,159,167,169,172,173,175,179,183,191,195,199,207,211,
%U A369375 223,239,255,286,295,313,316,319,335,337,351,367,373,383,406,415,417,433,435,447,455,461,463,479
%N A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).
%C A369375 Integers m > 0 such that 2^m-1 - 2^n is not prime for every natural n < m.
%C A369375 For m > 2, a number m is a term of this sequence if and only if A208083(m) = 0.
%C A369375 All Mersenne number m = 2^k-1 for k > 2 are in this sequence. The proof is below.
%C A369375 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.
%C A369375 If a = 2^m-1, then b < a, so for m > 2, 2^(2^m-1)-1 is a de Polignac number, QED.
%C A369375 Note that 2^(2^m-1)-1 - 2^n is divisible by some prime factor of 2^(2^m)-1.
%C A369375 Prime numbers of this sequence are Mersenne primes > 3, and many other primes.
%C A369375 Conjecture: if n > 5, then |2^(2^n-1)-1 - 2^m| is not prime for every m > 0.
%C A369375 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.
%C A369375 For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number.
%C A369375 It seems that the natural density of these numbers is about twice as high as the density of de Polignac numbers.
%C A369375 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).
%F A369375 For n > 1, a(n) = A138290(n-1) + 1.
%F A369375 A208083(a(n)) = 0, for n > 0.
%e A369375 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.
%e A369375 Note that both 2^148-1 and 2^148+1 are de Polignac numbers.
%t A369375 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 *)
%Y A369375 Cf. A000225, A006285, A000668, A101036, A138290, A208083, A232565, A369378.
%K A369375 nonn
%O A369375 1,2
%A A369375 _Thomas Ordowski_, Jan 22 2024
%E A369375 More terms from _Robert G. Wilson v_, Jan 22 2024