A347226 Safe primes (A005385) that are half-period primes (A097443).

83, 107, 227, 347, 359, 467, 479, 563, 587, 719, 839, 1187, 1283, 1307, 1319, 1439, 1523, 1907, 2027, 2039, 2879, 2963, 2999, 3119, 3203, 3467, 3803, 3947, 4079, 4283, 4547, 4679, 4787, 4799, 4919, 5387, 5399, 5483, 5507, 5639, 5879, 6599, 6719, 6827, 7079, 7187, 7523

Offset: 1

Author

Lamine Ngom, Aug 24 2021

Keywords

• nonn
• base

Comments

Apart from 5 and 11, a safe prime p is necessarily either a full reptend prime (A001913) or a half-period prime (A097443) since (p-1) is semiprime: 2*A005384(n) (Sophie Germain primes).

Safe primes being full reptend primes are listed in A000353.

a(n) is of the form 100*k + 10*{0, 2, 4, 6, 8} + {3, 7} or 100*k + 10*{1, 3, 5, 7, 9} + 9.

Number of terms < 10^k: 0, 1, 11, 56, 343, 2138, 15267, 114847, 886907, 7079602, ...

Examples

			(107-1)/2 = 53 is a prime, and the periodic part of the decimal expansion of 1/107 is of length 53.
Hence the safe prime 107 is in the sequence.
		

Crossrefs

Cf. A005385, A097443, A005384, A000353, A001913, A002371, A006883.

Programs

Maple

select(t -> isprime(t) and isprime((t - 1)/2) and numtheory:-order(10, t) = (t - 1)/2, [seq(t, t = 3 .. 10000, 2)]);

Mathematica

Select[Prime@Range@1000,PrimeQ[(#-1)/2]&&Length[First@@RealDigits[1/#]]==(#-1)/2&] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

Formula

A005385 INTERSECTION A097443.

a(n) == {17, 23, 29} mod 30.

a(n) == 11 (mod 12). - Hugo Pfoertner, Aug 24 2021