login
Primes of form n^2 - 3.
4

%I #60 Jan 12 2023 01:37:35

%S 13,61,97,193,397,673,1021,1153,1597,1933,2113,3361,4093,4621,6397,

%T 7393,7741,8461,9601,12097,12541,13921,15373,16381,18493,19597,20161,

%U 21313,26893,29581,36097,37633,40801,42433,43261,47521,48397

%N Primes of form n^2 - 3.

%C Also primes equal to the product of two consecutive odd numbers (A000466) minus 2. - _Giovanni Teofilatto_, Feb 11 2010

%C All terms are of the form 6m + 1. - _Zak Seidov_, May 01 2014

%H Nathaniel Johnston, <a href="/A028874/b028874.txt">Table of n, a(n) for n = 1..10000</a>

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>

%H R. J. Mathar, <a href="https://vixra.org/abs/2210.0029">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</a>, 2022.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>

%F A028872 INTERSECT A000040. - _Klaus Purath_, Dec 07 2020

%e 61 is prime and equal to 8^2 - 3, so it is in the sequence.

%e 67 is prime but it's 8^2 + 3 = 9^2 - 14, so it is not in the sequence.

%e 9^2 - 3 = 78 but it's composite, so it's not in the sequence either.

%t Select[Range[2, 250]^2 - 3, PrimeQ] (* _Harvey P. Dale_, Aug 07 2013 *)

%t Select[Table[n^2 - 3, {n, 2, 300}], PrimeQ] (* _Vincenzo Librandi_, Nov 08 2014 *)

%o (Magma) [a: n in [2..300] | IsPrime(a) where a is n^2-3 ]; // _Vincenzo Librandi_, Nov 08 2014

%o (PARI) select(isprime, vector(100,n,n^2-3)) \\ _Charles R Greathouse IV_, Nov 19 2014

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

%Y Primes terms in A082109. Subsequence of A068228. - _Klaus Purath_, Jan 09 2023

%K nonn

%O 1,1

%A _Patrick De Geest_