A222008 -id:A222008 - cp's OEIS frontend

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

Benoit Cloitre, Feb 12 2004

nonn

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

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.

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

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

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

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

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

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

A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

Original entry on oeis.org

29, 149, 269, 389, 509, 1109, 1229, 1949, 2309, 2909, 3989, 4349, 5189, 5309, 6269, 6389, 7109, 7949, 8069, 9749, 10589, 10709, 11069, 11549, 12149, 12269, 13229, 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

Offset: 1

Jonathan Sondow, Feb 02 2013

nonn

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.
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.
The corresponding primes p are A221982.
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
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

			29 is a member because 29 = 4*7 + 1 and 7 == 2 (mod 5) are prime.

P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F9, pp. 377-380.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, pp. 306-313.
Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004, revised 2012, p. 1.
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A001913, A005596, A006883, A045380, A106849, A221982, A222008.

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

Select[ Prime[ Range[4000]], Mod[(# - 1)/4, 5] == 2 && PrimeQ[(# - 1)/4] &]

is(n)=n%20==9 && isprime(n) && isprime(n\4) \\ Charles R Greathouse IV, Apr 18 2013

a(n) = 4*A221982(n) + 1.

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 30 2024

A378143 a(n) is the smallest prime of the form (2*p)^(2^n) + 1 for some prime p.

Original entry on oeis.org

5, 17, 257, 65537, 808551180810136214718004658177, 9807585394417153072393128067370344132933540474708183331242417216238928121991128579833857

Offset: 0

Juri-Stepan Gerasimov, Nov 17 2024

nonn

If p = 2, then a(n) is the Fermat prime.
Conjecture: the last digit of each value of a(n), where n >= 1, is 7.
The conjecture is equivalent to the claim that a(n) is not 10^(2^n) + 1 for any n, which in turn is equivalent to the claim that, if 10^(2^n) + 1 is prime, then either 4^(2^n) + 1 or 6^(2^n) + 1 is prime. - Charles R Greathouse IV, Nov 17 2024

Primes p such that (2*p)^(2^k) + 1 is prime: A005384 (k = 0), A052291 (k = 1), A378146 (k = 2).

Cf. A019434, A222008, A286678, A378134.

A221982 Primes p == 2 (mod 5) for which 4*p+1 is also prime.

Original entry on oeis.org

7, 37, 67, 97, 127, 277, 307, 487, 577, 727, 997, 1087, 1297, 1327, 1567, 1597, 1777, 1987, 2017, 2437, 2647, 2677, 2767, 2887, 3037, 3067, 3307, 3457, 3637, 3907, 4057, 4297, 4447, 4567, 4987, 5197, 5527, 5557, 6007, 6247, 6337, 6367, 6397, 6547, 6577, 7027, 7057, 7237, 7417, 7507, 7717, 7867

Offset: 1

Jonathan Sondow, Feb 02 2013

nonn

The corresponding primes 4*p+1 are Chebyshev's subsequence A221981 of the primes with primitive root 10.

			7 is a member because 7 == 2 (mod 5) and 29 = 4*7 + 1 are both prime.

P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.
R. K. Guy, Unsolved Problems in Number Theory, F9.

Paolo P. Lava, Table of n, a(n) for n = 1..10000
P. Moree, Artin's primitive root conjecture - a survey, arXiv 2004, revised 2012, p. 1.
Index entries for primes by primitive root

Cf. A001913, A006883, A045380, A106849, A221981, A222008.

A221982:=proc(q)
local n;
for n from 1 to q do
if isprime(n) and isprime(4*n+1) and (n mod 5)=2 then print(n) fi; od; end:
A221982 (10000); # Paolo P. Lava, Feb 12 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 2 && PrimeQ[4 # + 1] &]

a(n) = (A221981(n) - 1)/4.

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

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A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

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A378143 a(n) is the smallest prime of the form (2*p)^(2^n) + 1 for some prime p.

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A221982 Primes p == 2 (mod 5) for which 4*p+1 is also prime.

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Maple

Mathematica

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