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
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