A000353 -id:A000353 - cp's OEIS frontend

A000355 Primes p == 3, 9, 11 (mod 20) such that 2p+1 is also prime.

3, 11, 23, 29, 83, 89, 131, 191, 251, 431, 443, 491, 509, 683, 743, 809, 911, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1511, 1583, 1811, 1889, 1931, 2003, 2063, 2069, 2129, 2351, 2543, 2549, 2903, 2963, 2969, 3023, 3329, 3389, 3449, 3491, 3623, 3803

Offset: 1

Robert A. J. Matthews

a(n) = (A000353(n)-1)/2. - Reinhard Zumkeller, Feb 10 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Robert A. J. Matthews, Maximally periodic reciprocals, Bull. Institute of Mathematics and Its Applications, vol. 28, p. 147-148, 1992.

Subset of A005384.

Cf. A000353.

q:= p-> irem(p, 20) in {3, 9, 11} and andmap(isprime, [p,2*p+1]):
select(q, [$1..10000])[];  # Alois P. Heinz, Oct 31 2023

Select[Prime[Range[1000]], MatchQ[Mod[#, 20], 3|9|11] && PrimeQ[2#+1]&] (* Jean-François Alcover, Feb 07 2016 *)

is(n)=my(k=n%20); (k==3||k==9||k==11) && isprime(2*n+1) && isprime(n) \\ Charles R Greathouse IV, Nov 20 2014

More terms from Reinhard Zumkeller, Feb 10 2009

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

Original entry on oeis.org

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

Lamine Ngom, Aug 24 2021

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, ...

			(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.

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

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

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

A005385 INTERSECTION A097443.
a(n) == {17, 23, 29} mod 30.
a(n) == 11 (mod 12). - Hugo Pfoertner, Aug 24 2021

cp's OEIS Frontend

A000355 Primes p == 3, 9, 11 (mod 20) such that 2p+1 is also prime.

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