A393641 Primes p that are the greatest prime factor of 2^k + 1 for some k.

2, 3, 5, 11, 13, 17, 19, 41, 43, 109, 113, 241, 257, 331, 673, 683, 1321, 1613, 2113, 2731, 4051, 5419, 14449, 20857, 21841, 26317, 30269, 38737, 43691, 61681, 65537, 86171, 87211, 174763, 268501, 279073, 384773, 525313, 1564921, 2796203, 3033169, 4327489, 6700417

Offset

1,1

Comments

Numbers that occur in A002587.

2 and odd primes p such that k = A002326((p-1)/2) is even and 2^(k/2) + 1 is p-smooth.

Links

Table of n, a(n) for n=1..43.

Example

a(4) = 13 is a term because 13 is the greatest prime factor of 2^6 + 1 = 65 = 5 * 13.

Maple

A:= 2: count:= 1: P:= {}: p:= 2:
while count < 37 do
  p:= nextprime(p);
  m:= NumberTheory:-MultiplicativeOrder(2, p);
  if m::odd then next fi;
  m:= m/2;
  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 A019434.

Cf. A002326, A002587, A091317, A395945, A396002, A396044.

Sequence in context: A225184 A038947 A095315 * A221717 A140558 A387090

Adjacent sequences: A393638 A393639 A393640 * A393642 A393644 A393645

Keyword

nonn

Author

Robert Israel, May 13 2026

Extensions

a(40)-a(43) from Jinyuan Wang, May 23 2026

Status

approved