A028874 Primes of form k^2 - 3.

13, 61, 97, 193, 397, 673, 1021, 1153, 1597, 1933, 2113, 3361, 4093, 4621, 6397, 7393, 7741, 8461, 9601, 12097, 12541, 13921, 15373, 16381, 18493, 19597, 20161, 21313, 26893, 29581, 36097, 37633, 40801, 42433, 43261, 47521, 48397

Offset: 1

Author

Patrick De Geest

Keywords

• nonn

Comments

Also primes equal to the product of two consecutive odd numbers (A000466) minus 2. - Giovanni Teofilatto, Feb 11 2010

All terms are of the form 6m + 1. - Zak Seidov, May 01 2014

Examples

			61 is prime and equal to 8^2 - 3, so it is in the sequence.
67 is prime but it's 8^2 + 3 = 9^2 - 14, so it is not in the sequence.
9^2 - 3 = 78 but it's composite, so it's not in the sequence either.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
• R. J. Mathar, Solutions to the exponential Diophantine 1 + p_1^x + p_2^y + p_3^z = w^2 for distinct primes p_1, p_2, p_3, 2022.
• Eric Weisstein's World of Mathematics, Near-Square Prime.

Crossrefs

Cf. A002476 (Primes of form 6m + 1), A028871, A028872, A028873.

Primes terms in A082109. Subsequence of A068228. - Klaus Purath, Jan 09 2023

Programs

Magma

[a: n in [2..300] | IsPrime(a) where a is n^2-3 ]; // Vincenzo Librandi, Nov 08 2014

Mathematica

Select[Range[2, 250]^2 - 3, PrimeQ] (* Harvey P. Dale, Aug 07 2013 *)
Select[Table[n^2 - 3, {n, 2, 300}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *)

PARI

select(isprime, vector(100,n,n^2-3)) \\ Charles R Greathouse IV, Nov 19 2014

Formula

A028872 INTERSECT A000040. - Klaus Purath, Dec 07 2020

a(n) = A028873(n)^2 - 3. - Amiram Eldar, Mar 01 2025