A090866 Primes p == 1 (mod 4) such that (p-1)/4 is prime.

13, 29, 53, 149, 173, 269, 293, 317, 389, 509, 557, 653, 773, 797, 1109, 1229, 1493, 1637, 1733, 1949, 1997, 2309, 2477, 2693, 2837, 2909, 2957, 3413, 3533, 3677, 3989, 4133, 4157, 4253, 4349, 4373, 4493, 4517, 5189, 5309, 5693, 5717, 5813, 6173, 6197

Offset: 1

Author

Benoit Cloitre, Feb 12 2004

Keywords

• nonn

Comments

Same as Chebyshev's subsequence of the primes with primitive root 2, because Chebyshev showed that 2 is a primitive root of all primes p = 4*q+1 with q prime. If the sequence is infinite, then Artin's conjecture ("every nonsquare positive integer n is a primitive root of infinitely many primes q") is true for n = 2. - Jonathan Sondow, Feb 04 2013

References

• Albert H. Beiler: Recreations in the theory of numbers. New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.
• P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to Artin's conjecture

Crossrefs

Cf. A001122, A005385, A005596, A023212, A221981, A222008.

Programs

Magma

f:=[n: n in [1..2000] | IsPrime(n) and IsPrime(4*n+1)]; [4*f[n] + 1: n in [1..50]]; // G. C. Greubel, Feb 08 2019

Mathematica

Select[Prime[Range[1000]], Mod[#, 4]==1 && PrimeQ[(#-1)/4] &] (* G. C. Greubel, Feb 08 2019 *)

PARI

isok(p) = isprime(p) && !frac(q=(p-1)/4) && isprime(q); \\ Michel Marcus, Feb 09 2019

Formula

a(n) = 4*A023212(n) + 1.