A121259 Numbers k such that (3*k^2 + 1)/4 is prime.

3, 5, 7, 9, 13, 19, 21, 23, 27, 29, 35, 47, 49, 51, 55, 57, 61, 65, 69, 75, 77, 83, 85, 91, 97, 99, 105, 111, 117, 125, 127, 133, 135, 149, 161, 163, 173, 177, 181, 183, 187, 191, 211, 217, 239, 247, 251, 257, 259, 273, 281, 285, 295, 307, 313, 315, 317, 329, 331, 341

Offset: 1

Author

Zak Seidov, Aug 23 2006

Keywords

• nonn
• easy

Examples

			(3*5^2 + 1)/4 = 19 is the 2nd prime of this form, so a(2) = 5.
(3*13^2 + 1)/4 = 127 is the 5th prime of this form, so a(5) = 13.
(3*19^2 + 1)/4 = 271 is the 6th prime of this form, so a(6) = 19.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..8144

Crossrefs

Cf. comment by Michael Somos in A002407.

Cf. A002504, A111251, A111051 (simpler variant).

Programs

Mathematica

Select[Range[400],PrimeQ[(3#^2+1)/4]&]  (* Harvey P. Dale, Mar 24 2011 *)

PARI

is(n) = my(p=(3*n^2+1)/4); (denominator(p)==1) && isprime(p); \\ Charles R Greathouse IV, Jun 13 2017; edited by Michel Marcus, Oct 12 2023

Formula

a(n) = sqrt((4*A002407(n) - 1)/3). [corrected by Rémi Guillaume, Dec 07 2023]

a(n) = 2*A002504(n) - 1. - Hugo Pfoertner, Oct 07 2023

a(n) = 2*A111251(n) + 1. - Rémi Guillaume, Dec 06 2023