A162857 Primes of the form 4p - 1, p a prime.

7, 11, 19, 43, 67, 163, 211, 283, 331, 523, 547, 691, 787, 907, 1051, 1123, 1171, 1531, 1723, 1867, 2011, 2083, 2251, 2347, 2371, 2467, 2707, 2731, 2803, 2971, 3187, 3307, 3547, 3643, 3907, 3931, 4051, 4243, 4363, 4603, 4651, 4723, 5107, 5227, 5443, 5923

Offset

1,1

Comments

If 4p - 1 is prime then n^2 + n + p = p(4p - 1) for some n = 1, 2, 3, ... [Proof. Let n + 1 = 2p, etc.]

From Alonso del Arte, Jan 14 2024: (Start)

The first six terms correspond to rings of algebraic integers of Q(sqrt(-a(n))) which are unique factorization domains.

In the ring of algebraic integers of Q(sqrt(-a(n))), the corresponding prime p = (a(n) + 1)/4 is divisible by 1/2 - sqrt(-a(n))/2 and 1/2 + sqrt(-a(n))/2, both of those being algebraic integers with minimal polynomial x^2 - x + p. For example, in Q(sqrt(-163)), we see that (1/2 - sqrt(-163)/2)(1/2 + sqrt(-163)/2) = 1/4 + 163/4 = 41, with both of the divisors having the minimal polynomial x^2 - x + 41. (End)

Links

Table of n, a(n) for n=1..46.

Formula

a(n) >> n log^2 n. - Charles R Greathouse IV, Jun 14 2022

Mathematica

Select[Prime[Range[1000]], PrimeQ[(# + 1)/4] &] (* Alonso del Arte, Jan 14 2024 *)

Prog

(PARI) is(n)=isprime(n) && isprime(4*n-1) \\ Charles R Greathouse IV, Jun 14 2022
(Scala) def isPrime(num: Int): Boolean = Math.abs(num) match {
  case 0 => false; case 1 => false; case n => (2 to Math.floor(Math.sqrt(n)).toInt) forall (p => n % p != 0)
}
((1 to 2500).map(4 * _ - 1)).filter(n => isPrime(n) && isPrime((n + 1)/4)) // Alonso del Arte, Jan 14 2024

Crossrefs

Cf. A062737 for the corresponding primes p.

Overlaps with A003173, the Heegner numbers (last six terms of that one match the first six of this one).

Sequence in context: A055939 A154555 A285016 * A307965 A323109 A023267

Adjacent sequences: A162854 A162855 A162856 * A162858 A162859 A162860

Keyword

nonn,easy

Author

Daniel Tisdale, Jul 14 2009

Extensions

More terms from N. J. A. Sloane, Jul 19 2009

Status

approved