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A080020 Primes of the form 9k^2 + 3k + 367, where k can be negative. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 08:36 EST 2021. Contains 349543 sequences. (Running on oeis4.)