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

1,1

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, ...

Links

Table of n, a(n) for n=1..47.

Formula

A005385 INTERSECTION A097443.

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

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

Example

(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.

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 *)

Crossrefs

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

Sequence in context: A139765 A031412 A033252 * A261449 A288880 A126117

Adjacent sequences: A347223 A347224 A347225 * A347227 A347228 A347229

Keyword

nonn,base

Author

Lamine Ngom, Aug 24 2021

Status

approved