A395945 Primes p that are the greatest prime factor of 2^k - 1 for some k >= 1.

3, 5, 7, 13, 17, 31, 41, 73, 89, 109, 127, 151, 241, 257, 331, 337, 673, 683, 1321, 1801, 2089, 2113, 4051, 5419, 8191, 14449, 23311, 38737, 61681, 65537, 121369, 122921, 131071, 152041, 178481, 201961, 262657, 268501, 279073, 524287, 525313

Offset

1,1

Comments

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

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

Links

Table of n, a(n) for n=1..41.

Example

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

Maple

A:= NULL: count:= 0: P:= {}: p:= 2:
while count < 40 do
  p:= nextprime(p);
  m:= NumberTheory:-MultiplicativeOrder(2, p);
  t:= 2^m-1;
  t:= t/p^padic:-ordp(t, p);
  for q in P do if t mod q = 0 then
    t:= t/q^padic:-ordp(t, q);
    if t = 1 then break fi;
  fi od;
  if t = 1 then count:= count+1; A:= A, p fi;
  P:= P union {p};
od:
A;

Crossrefs

Includes A000668. Cf. A002326, A006530, A395931.

Sequence in context: A339506 A178490 A182981 * A234388 A003424 A073638

Adjacent sequences: A395942 A395943 A395944 * A395946 A395947 A395948

Keyword

nonn

Author

Robert Israel, May 11 2026

Status

approved