A065406 Mersenne prime exponents (A000043) which are also Sophie Germain primes (A005384).

2, 3, 5, 89, 9689, 21701, 859433, 43112609

Offset

1,1

Comments

From Gord Palameta, Jul 19 2018: (Start)

All terms after the first two are congruent to 1 modulo 4, because if p is a Sophie Germain prime that is congruent to 3 modulo 4 then 2p + 1 divides 2^p - 1.

Boklan and Conway conjecture that this sequence is finite.

(End)

Links

Table of n, a(n) for n=1..8.

David W. Ash, Ian F. Blake, and Scott A. Vanstone, Low Complexity Normal Bases, Discrete Applied Mathematics, 25(1989), 206.

Kent D. Boklan, John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016, p. 6.

Example

31 = 2^5 - 1 and 11 = 2 * 5 + 1 are primes.

Mathematica

Select[Prime[Range[1000]], PrimeQ[2# + 1] && PrimeQ[2^# - 1] &] (* Alonso del Arte, Jul 20 2018 *)
Select[Prime@ Range[10^6], And[PrimeQ[2 # + 1], MersennePrimeExponentQ@ #] &] (* Michael De Vlieger, Jul 20 2018 *)

Crossrefs

Cf. A000043, A005384.

Sequence in context: A208226 A259378 A155011 * A111331 A205668 A118505

Adjacent sequences: A065403 A065404 A065405 * A065407 A065408 A065409

Keyword

nonn,more

Author

Labos Elemer, Nov 06 2001

Extensions

a(8) = 43112609, since the ordinal position of this term in A000043 is now confirmed. - Gord Palameta, Jul 19 2018

Status

approved