A395945 - OEIS

%I #9 May 18 2026 10:13:02

%S 3,5,7,13,17,31,41,73,89,109,127,151,241,257,331,337,673,683,1321,

%T 1801,2089,2113,4051,5419,8191,14449,23311,38737,61681,65537,121369,

%U 122921,131071,152041,178481,201961,262657,268501,279073,524287,525313

%N Primes p that are the greatest prime factor of 2^k - 1 for some k >= 1.

%C Solutions of p = A006530(2^(A002326((p-1)/2))-1).

%C Odd primes p such that if k is the multiplicative order of 2 mod p, 2^k - 1 is p-smooth.

%e a(9) = 89 is a term because 2^11 - 1 = 23 * 89 whose greatest prime factor is 89.

%p A:= NULL: count:= 0: P:= {}: p:= 2:

%p while count < 40 do

%p   p:= nextprime(p);

%p   m:= NumberTheory:-MultiplicativeOrder(2,p);

%p   t:= 2^m-1;

%p   t:= t/p^padic:-ordp(t,p);

%p   for q in P do if t mod q = 0 then

%p     t:= t/q^padic:-ordp(t,q);

%p     if t = 1 then break fi;

%p   fi od;

%p   if t = 1 then count:= count+1; A:= A,p fi;

%p   P:= P union {p};

%p od:

%p A;

%Y Includes A000668.  Cf. A002326, A006530, A395931.

%K nonn

%O 1,1

%A _Robert Israel_, May 11 2026