This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A221981 #50 Jan 29 2025 13:12:35
%S A221981 29,149,269,389,509,1109,1229,1949,2309,2909,3989,4349,5189,5309,6269,
%T A221981 6389,7109,7949,8069,9749,10589,10709,11069,11549,12149,12269,13229,
%U A221981 13829,14549,15629,16229,17189,17789,18269,19949,20789,22109,22229,24029,24989,25349,25469,25589,26189,26309,28109,28229,28949,29669,30029,30869,31469,32069,33149,34589,34949,36269,36629,36749,37589
%N A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.
%C A221981 Moree (2012) says that Chebyshev observed that if q = 4p + 1 is prime, with prime p == 2 (mod 5), then 10 is a primitive root modulo q.
%C A221981 If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 10.
%C A221981 The corresponding primes p are A221982.
%C A221981 The sequence is infinite under Dickson's conjecture, thus Dickson's conjecture implies Artin's conjecture for n = 10. - _Charles R Greathouse IV_, Apr 18 2013
%C A221981 Two conjectures: (a) These primes have primitive root 40; (b) if a(n)*8 + 1 is prime then it has primitive root 10. - _Davide Rotondo_, Dec 31 2024
%D A221981 P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.
%D A221981 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F9, pp. 377-380.
%H A221981 Amiram Eldar, <a href="/A221981/b221981.txt">Table of n, a(n) for n = 1..10000</a>
%H A221981 P. L. Chebyshev, <a href="http://archive.org/stream/theoriedercongr00schagoog#page/n330/mode/2up">Theorie der Congruenzen</a>, Mayer & Mueller, 1889, pp. 306-313.
%H A221981 Pieter Moree, <a href="http://arxiv.org/abs/math/0412262">Artin's primitive root conjecture - a survey</a>, arXiv:math/0412262 [math.NT], 2004, revised 2012, p. 1.
%H A221981 <a href="/index/Ar#Artin">Index entries for sequences related to Artin's conjecture</a>
%H A221981 <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>
%F A221981 a(n) = 4*A221982(n) + 1.
%F A221981 a(n) >> n log^2 n. - _Charles R Greathouse IV_, Dec 30 2024
%e A221981 29 is a member because 29 = 4*7 + 1 and 7 == 2 (mod 5) are prime.
%p A221981 A221981:=n->`if`(isprime(4*n+1) and isprime(n) and n mod 5 = 2, 4*n+1, NULL): seq(A221981(n), n=1..10^4); # _Wesley Ivan Hurt_, Dec 11 2015
%t A221981 Select[ Prime[ Range[4000]], Mod[(# - 1)/4, 5] == 2 && PrimeQ[(# - 1)/4] &]
%o A221981 (PARI) is(n)=n%20==9 && isprime(n) && isprime(n\4) \\ _Charles R Greathouse IV_, Apr 18 2013
%Y A221981 Cf. A001913, A005596, A006883, A045380, A106849, A221982, A222008.
%K A221981 nonn
%O A221981 1,1
%A A221981 _Jonathan Sondow_, Feb 02 2013