cp's OEIS Frontend

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

%I A080020 #26 Jun 11 2021 19:03:57
%S A080020 367,373,379,397,409,439,457,499,523,577,607,673,709,787,829,919,967,
%T A080020 1069,1123,1237,1297,1423,1489,1627,1699,2089,2347,2437,2719,2917,
%U A080020 3019,3229,3559,3673,3907,4027,4273,4657,4789,5059,5197,5479,5623,6067,6373
%N A080020 Primes of the form 9k^2 + 3k + 367, where k can be negative.
%C A080020 Original definition: Primes of the form q(n) = 370 + 18*binomial(ceiling(n/2), 2) + 3*(-1)^n*(2*ceiling(n/2) - 1).
%C A080020 The smallest positive k for which q(k) is not prime is k = 26.
%C A080020 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).
%C A080020 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
%H A080020 Charles R Greathouse IV, <a href="/A080020/b080020.txt">Table of n, a(n) for n = 1..10000</a>
%F A080020 From _Alonso del Arte_, Nov 16 2017: (Start)
%F A080020 ((6n - 1)^2 + 1467)/4 = (36n^2 - 12n + 1468)/2 = 9n^2 - 3n + 367.
%F A080020 ((6n + 1)^2 + 1467)/4 = (36n^2 + 12n + 1468)/2 = 9n^2 + 3n + 367. (End)
%e A080020 Given k = -2, we have 9 * 4 - 3 * 2 + 367 = 36 - 6 + 367 = 397 (a prime).
%e A080020 Given k = -1, we have 9 - 3 + 367 = 373 (a prime).
%e A080020 Given k = 0, we have 367 (a prime).
%e A080020 Given k = 1, we have 9 + 3 + 367 = 379 (a prime).
%e A080020 Given k = 2, we have 9 * 4 + 3 * 2 + 367 = 36 + 6 + 367 = 409 (a prime).
%t A080020 Union[Select[Table[9n^2 + 3n + 367, {n, -30, 30}], PrimeQ]] (* _Harvey P. Dale_, Mar 23 2013 *)
%Y A080020 Cf. A005846.
%K A080020 nonn,easy
%O A080020 1,1
%A A080020 _T. Amdeberhan_, Jan 20 2003
%E A080020 Edited by _Dean Hickerson_, Jan 20 2003
%E A080020 New definition from _Charles R Greathouse IV_, Feb 15 2011