A179113 - OEIS

%I #26 Aug 08 2019 01:57:55

%S 31,89,127,223,233,431,601,881,911,1103,1801,2089,2351,3191,3391,4513,

%T 5209,6361,8191,9623,9719,11447,11471,13367,14951,15193,15809,18041,

%U 18121,18199,18287,20231,23279,23671,39551,43441,50023,53993,54217,55441,55871,59233

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

%C 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.

%C 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

%H Robert Israel, <a href="/A179113/b179113.txt">Table of n, a(n) for n = 1..129</a>

%H Mariusz Skałba, <a href="https://doi.org/10.1007/s00017-004-0211-x">Two conjectures on primes dividing 2^a+2^b+1</a>, Elemente der Mathematik 59 (2004), issue 4, pp. 171-173.

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

%p N:= 10000; # to test the first N primes for membership

%p A179113:= proc(p)

%p           local x, R;

%p x:= 1; R:= {};

%p do

%p   R:= R union {p-1-x};

%p   if member(x,R) then return(false) end if;

%p   x:= 2*x mod p;

%p   if x = 1 then return(true) end if;

%p end do;

%p end proc;

%p select(A179113,[seq(ithprime(i),i=2..N)]);

%p # _Robert Israel_, May 19 2013

%t 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 *)

%o (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", ")))

%K nonn

%O 1,1

%A _Phil Carmody_, Jan 04 2011

%E More terms from _Robert Israel_, May 19 2013