A179113 Odd primes which can never divide 2^a+2^b+1.

31, 89, 127, 223, 233, 431, 601, 881, 911, 1103, 1801, 2089, 2351, 3191, 3391, 4513, 5209, 6361, 8191, 9623, 9719, 11447, 11471, 13367, 14951, 15193, 15809, 18041, 18121, 18199, 18287, 20231, 23279, 23671, 39551, 43441, 50023, 53993, 54217, 55441, 55871, 59233

Offset

1,1

Comments

Contains the Mersenne primes M_p for p>3 as a subsequence, as 2^a+2^b cannot exceed 2^(p-1)+2^(p-2) which is less than 2^p-2 is p>3.

Mariusz Skałba conjectures that this sequence has density zero among all primes but contains infinitely many primes based on the following observations. For any prime p in this sequence, the multiplicative order of 2 modulo p is <p^0.8 (Erdős conjectures that the set of such primes must have density zero among all primes). Moreover, any number of the form 2^m-1 the number of whose prime factors counted with multiplicity is <log m/log 3 has at least one prime factor in this sequence (the infinitude of such numbers may be more tractable than the infinitude of Mersenne primes). - Tomohiro Yamada, Aug 08 2019

Links

Robert Israel, Table of n, a(n) for n = 1..129

Mariusz Skałba, Two conjectures on primes dividing 2^a+2^b+1, Elemente der Mathematik 59 (2004), issue 4, pp. 171-173.

Example

31 is on the list as you can't sum any two of {1, 2, 4, 8, 16} to make 30 (mod 31).

Maple

N:= 10000; # to test the first N primes for membership
A179113:= proc(p)
          local x, R;
x:= 1; R:= {};
do
  R:= R union {p-1-x};
  if member(x, R) then return(false) end if;
  x:= 2*x mod p;
  if x = 1 then return(true) end if;
end do;
end proc;
select(A179113, [seq(ithprime(i), i=2..N)]);
# Robert Israel, May 19 2013

Mathematica

n = 10000; (* to test the first n primes for membership *) A179113[p_] := Module[{x = 1, r = {}}, While[True, r = r ~Union~ {p-1-x}; If[MemberQ[r, x], Return[False]]; x = Mod[2*x, p]; If[x == 1, Return[True]]]]; Reap[Do[If[A179113[p], Print[p]; Sow[p]], {p, Prime /@ Range[2, n]}]][[2, 1]] (* Jean-François Alcover, Dec 02 2013, translated from Robert Israel's Maple program *)

Prog

(PARI) forprime(p=3, 1000, pol=x+O(x^p); t=2; while(t-1, pol+=x^t; t=t*2%p); pol2=pol*pol; if(!polcoeff(pol2, p-1), print1(p", ")))

Crossrefs

Sequence in context: A005184 A096731 A039518 * A142715 A093758 A359650

Adjacent sequences: A179110 A179111 A179112 * A179114 A179115 A179116

Keyword

nonn

Author

Phil Carmody, Jan 04 2011

Extensions

More terms from Robert Israel, May 19 2013

Status

approved