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%I #32 Sep 08 2022 08:45:12
%S 13,29,53,149,173,269,293,317,389,509,557,653,773,797,1109,1229,1493,
%T 1637,1733,1949,1997,2309,2477,2693,2837,2909,2957,3413,3533,3677,
%U 3989,4133,4157,4253,4349,4373,4493,4517,5189,5309,5693,5717,5813,6173,6197
%N Primes p == 1 (mod 4) such that (p-1)/4 is prime.
%C 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
%D Albert H. Beiler: Recreations in the theory of numbers. New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.
%D P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.
%H G. C. Greubel, <a href="/A090866/b090866.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Ar#Artin">Index entries for sequences related to Artin's conjecture</a>
%F a(n) = 4*A023212(n) + 1.
%t Select[Prime[Range[1000]], Mod[#, 4]==1 && PrimeQ[(#-1)/4] &] (* _G. C. Greubel_, Feb 08 2019 *)
%o (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
%o (PARI) isok(p) = isprime(p) && !frac(q=(p-1)/4) && isprime(q); \\ _Michel Marcus_, Feb 09 2019
%Y Cf. A001122, A005385, A005596, A023212, A221981, A222008.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Feb 12 2004
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