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Primes of form 4*n^2 + 4*n + 653.
4

%I #16 Sep 08 2022 08:45:38

%S 653,661,677,701,733,773,821,877,941,1013,1093,1181,1277,1381,1493,

%T 1613,1741,1877,2333,2677,2861,3253,3461,3677,4133,4373,4621,4877,

%U 5413,5693,5981,6277,6581,7213,7541,7877,8221,8573,8933,9677,10061,10453,10853

%N Primes of form 4*n^2 + 4*n + 653.

%C First 18 terms are for n from 0 through 17, next terms are for n = 20, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, ...

%C The sequence of n such that 4*n^2 + 4*n + 653 is composite starts 18, 19, 21, 24, 28, 33, 39, 46, 54, 60, 61, 62, 63, 65, 67, 72, 73, 75, 81, 82, 84, 85, 86, 93, 95, 96, 100, ...

%C These primes are in A000414. [_Bruno Berselli_, Apr 20 2014]

%H Vincenzo Librandi, <a href="/A145202/b145202.txt">Table of n, a(n) for n = 1..1000</a>

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

%e a(18) = 4*17^2 + 4*17 + 653 = 1877.

%t Select[Table[4 n^2 + 4 n + 653, {n, 0, 100}], PrimeQ] (* _Vincenzo Librandi_, Apr 21 2014 *)

%o (PARI) {for(n=0, 50, if(isprime(p=4*n^2+4*n+653), print1(p, ",")))}

%o (Magma) [a: n in [0..100] | IsPrime(a) where a is 4*n^2 + 4*n + 653]; // _Vincenzo Librandi_, Apr 21 2014

%Y A145125 is essentially the same sequence.

%Y Cf. A005846 (primes of form n^2 + n + 41).

%K nonn

%O 1,1

%A _Klaus Brockhaus_, Oct 04 2008