A281022 Single (or isolated or non-twin) primes that are also safe primes.

23, 47, 83, 167, 263, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1187, 1283, 1307, 1367, 1439, 1523, 1823, 1907, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903, 2963, 3023, 3203, 3623, 3779, 3803, 3863, 3947, 4007, 4079, 4139, 4283, 4679, 4703, 4919

Offset: 1

Author

Altug Alkan, Jan 13 2017

Keywords

• nonn
• easy

Comments

Primes p such that neither p - 2 nor p + 2 is prime while (p - 1) / 2 is prime.

It is conjectured that there are infinitely many safe primes, but this is still unproved, so it is not known whether this sequence is infinite.

Examples

			23 is a term because 23 - 2 = 21 and 23 + 2 = 25 are composite and (23 - 1) / 2 = 11 is prime.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005385, A007510, A230117.

Programs

Mathematica

Select[Prime[Range[700]],Boole[PrimeQ[{#+2,#-2,(#-1)/2}]]=={0,0,1}&] (* Harvey P. Dale, Aug 14 2023 *)

PARI

lista(nn) = { forprime(p=11, nn, if(!isprime(p+2) && isprime((p-1)/2), print1(p, ", ")));}

Formula

a(n) = 2 * A230117(n+1) + 1, for n > 0.