A080020 Primes of the form 9k^2 + 3k + 367, where k can be negative.

367, 373, 379, 397, 409, 439, 457, 499, 523, 577, 607, 673, 709, 787, 829, 919, 967, 1069, 1123, 1237, 1297, 1423, 1489, 1627, 1699, 2089, 2347, 2437, 2719, 2917, 3019, 3229, 3559, 3673, 3907, 4027, 4273, 4657, 4789, 5059, 5197, 5479, 5623, 6067, 6373

Offset

1,1

Comments

Original definition: Primes of the form q(n) = 370 + 18*binomial(ceiling(n/2), 2) + 3*(-1)^n*(2*ceiling(n/2) - 1).

The smallest positive k for which q(k) is not prime is k = 26.

Every q(k) is a divisor of some value of e(x) = x^2 + x + 41, the Euler prime-generating polynomial. Specifically, e(3*k^2 - 2*k + 122) = q(2*k) * e(k-1) and e(3*k^2 + 2*k + 122) = q(2*k + 1) * e(k).

Also primes of the form (k^2 + 1467)/4 with k odd. These primes are composite in O_Q(sqrt(-163)), since they can be expressed as (k/2 - 3*sqrt(-163))*(k/2 + 3*sqrt(-163)). For example, (7/2 - 3*sqrt(-163)/2)(7/2 + 3*sqrt(-163)/2) = 379. - Alonso del Arte, Nov 15 2017

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

From Alonso del Arte, Nov 16 2017: (Start)

((6n - 1)^2 + 1467)/4 = (36n^2 - 12n + 1468)/2 = 9n^2 - 3n + 367.

((6n + 1)^2 + 1467)/4 = (36n^2 + 12n + 1468)/2 = 9n^2 + 3n + 367. (End)

Example

Given k = -2, we have 9 * 4 - 3 * 2 + 367 = 36 - 6 + 367 = 397 (a prime).
Given k = -1, we have 9 - 3 + 367 = 373 (a prime).
Given k = 0, we have 367 (a prime).
Given k = 1, we have 9 + 3 + 367 = 379 (a prime).
Given k = 2, we have 9 * 4 + 3 * 2 + 367 = 36 + 6 + 367 = 409 (a prime).

Mathematica

Union[Select[Table[9n^2 + 3n + 367, {n, -30, 30}], PrimeQ]] (* Harvey P. Dale, Mar 23 2013 *)

Crossrefs

Cf. A005846.

Sequence in context: A279982 A286793 A079493 * A192448 A118566 A320711

Adjacent sequences: A080017 A080018 A080019 * A080021 A080022 A080023

Keyword

nonn,easy

Author

T. Amdeberhan, Jan 20 2003

Extensions

Edited by Dean Hickerson, Jan 20 2003

New definition from Charles R Greathouse IV, Feb 15 2011

Status

approved