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

Author

T. Amdeberhan, Jan 20 2003

Keywords

• nonn
• easy

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A005846.

Programs

Mathematica

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

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)

Extensions

Edited by Dean Hickerson, Jan 20 2003

New definition from Charles R Greathouse IV, Feb 15 2011