A221981 -id:A221981 - 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.

A222008 Primes of the form 4^k + 1 for some k > 0.

Original entry on oeis.org

5, 17, 257, 65537

Offset: 1

Jonathan Sondow, Feb 04 2013

nonn

Same as Fermat primes 2^(2^m) + 1 for m >= 1. See A019434 for comments, etc.
Chebyshev showed that 3 is a primitive root of all primes of the form 2^(2*k) + 1 with k > 0. 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 = 3.
As a(n) is a Fermat prime > 3, by Pépin's test a(n) has primitive root 3.
Conjecture: let p a prime number, a(n) is not congruent to p mod (p^2-3)/2. - Vincenzo Librandi, Jun 15 2014
This conjecture is false when p = a(n), but may be true for primes p != a(n). - Jonathan Sondow, Jun 15 2014
Primes p with the property that k-th smallest divisor of its squares p^2, for all 1 <= k <= tau(p^2), contains exactly k "1" digits in its binary representation. Corresponding values of squares p^2: 25, 289, 66049, 4295098369. Example: p = 257, set of divisors of p^2 = 66049 in binary representation: {1, 100000001, 10000001000000001}. Subsequence of A255401. - Jaroslav Krizek, Dec 21 2016

			4^1 + 1 = 5 is prime, so a(1) = 5. Also, 3^k == 3, 4, 2, 1 (mod 5) for k = 1, 2, 3, 4, resp., so 3 is a primitive root for a(1).

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 102, nr. 3.
P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.

P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, p. 306-313.
R. Fueter, Über primitive Wurzeln von Primzahlen, Comment. Math. Helv., 18 (1946), 217-223, p. 217.
Wikipedia, Pépin's test
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A005596, A019334, A019434, A090866, A221981.

Select[Table[4^k + 1, {k, 10^3}], PrimeQ] (* Michael De Vlieger, Dec 22 2016 *)

a(n) = A019434(n+1) for n > 0.

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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A222008 Primes of the form 4^k + 1 for some k > 0.

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

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