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

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

Offset: 1

Author

Labos Elemer, Nov 06 2001

Keywords

• nonn
• more

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)

Examples

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

Links

• 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.

Crossrefs

Cf. A000043, A005384.

Programs

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 *)

Extensions

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