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%I #21 Sep 08 2022 08:45:46
%S 3,19,31,83,131,223,383,479,643,1279,1823,2131,2239,2579,2819,3331,
%T 4483,4639,6163,6719,7103,7699,8963,9631,9859,10559,11779,13331,14143,
%U 14419,15263,17939,19843,21503,22531,24659,25759,28031,29599,30803,35423
%N Primes of the form k*(k+2)/3 - 2, k > 0.
%C Or: primes of the form k*(k+1)*(k+2)/(k+(k+1)+(k+2))-2.
%C Generated by k=3, 7, 9, 15, 19, 25, 33, 37, 43, ....
%C Primes p such that 3*p+7 is a square. - _Vincenzo Librandi_, Dec 05 2015
%C Primes of the forms 3*k^2 + 2*k - 2 and 3*k^2 + 4*k - 1. - _Robert Israel_, Nov 27 2017
%H Robert Israel, <a href="/A162307/b162307.txt">Table of n, a(n) for n = 1..10000</a>
%e k=3 contributes a term because 3*(3+2)/3 - 2 = 3 = a(1) is prime.
%p select(isprime, [seq(seq((3*j+i)*(3*j+i+2)/3-2, i=0..1),j=1..1000)]); # _Robert Israel_, Nov 27 2017
%t f[n_]:=(n*(n+1)*(n+2))/(n+(n+1)+(n+2))-2; lst={};Do[p=f[n];If[PrimeQ[p], AppendTo[lst,p]],{n,6!}];lst
%t Select[Table[(k(k+2))/3-2,{k,350}],PrimeQ] (* _Harvey P. Dale_, May 10 2014 *)
%o (Magma) [p: p in PrimesUpTo(45000) | IsSquare(3*p+7)]; // _Vincenzo Librandi_, Dec 05 2015
%o (PARI) forprime(p=2, 1e5, if(issquare(3*p+7), print1(p , ", "))) \\ _Altug Alkan_, Dec 05 2015
%K nonn,easy
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Jun 30 2009
%E Definition simplified by _R. J. Mathar_, Jul 02 2009
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