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%I #24 Sep 08 2022 08:45:12
%S 3,19,73,163,883,1153,1459,1801,2179,2593,3529,4051,8713,10369,11251,
%T 15139,17299,18433,19603,20809,22051,30259,34849,36451,46819,48673,
%U 52489,62659,69193,71443,80803,83233,95923,103969,112339,115201,130051
%N Primes of the form 2*n^2+1.
%C A prime p can be expressed as either the sum of two squares or the sum of two squares - 1, p = X^2 + Y^2 or p = X^2 + Y^2 - 1, if and only if p is of the form 2*(m^2)+1 where m is either 1 or a multiple of 3.
%C Conjecture: 2^(a(n)-1) - 3 is not prime. - _Vincenzo Librandi_, Feb 04 2013.
%C Primes in A058331. - _Vincenzo Librandi_, Apr 10 2015
%H Vincenzo Librandi, <a href="/A090698/b090698.txt">Table of n, a(n) for n = 1..5000</a>
%F a(n)=2*A089001(n)^2+1 = A000040(A090612(n)).
%e 19 = 2^2 + 4^2 - 1 = 2*(3^2)+1
%e 73 = 5^2 + 7^2 - 1 = 2*(6^2)+1
%e 163= 8^2 + 10^2 -1 = 2*(9^2)+1
%e 883= 10^2+ 28^2 -1 = 2*(21^2)+1
%t Select[Table[2n^2+1,{n,0,900}],PrimeQ] (* _Vincenzo Librandi_, Dec 02 2011 *)
%o (Magma)[a: n in [0..400] | IsPrime(a) where a is 2*n^2+1];// _Vincenzo Librandi_, Dec 02 2011
%o (PARI) is(n)=isprime(2*n^2+1) \\ _Charles R Greathouse IV_, Jan 05 2013
%Y Cf. A058331, A089001, A089008, A090612.
%K nonn,easy
%O 1,1
%A _Kurmang. Aziz. Rashid_, Dec 20 2003
%E Extended by _Ray Chandler_, Dec 21 2003
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