A090698 Primes of the form 2*n^2+1.

3, 19, 73, 163, 883, 1153, 1459, 1801, 2179, 2593, 3529, 4051, 8713, 10369, 11251, 15139, 17299, 18433, 19603, 20809, 22051, 30259, 34849, 36451, 46819, 48673, 52489, 62659, 69193, 71443, 80803, 83233, 95923, 103969, 112339, 115201, 130051

Offset: 1

Author

Kurmang. Aziz. Rashid, Dec 20 2003

Keywords

• nonn
• easy

Comments

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.

Conjecture: 2^(a(n)-1) - 3 is not prime. - Vincenzo Librandi, Feb 04 2013.

Primes in A058331. - Vincenzo Librandi, Apr 10 2015

Examples

			19 = 2^2 + 4^2 - 1 = 2*(3^2)+1
73 = 5^2 + 7^2 - 1 = 2*(6^2)+1
163= 8^2 + 10^2 -1 = 2*(9^2)+1
883= 10^2+ 28^2 -1 = 2*(21^2)+1
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A058331, A089001, A089008, A090612.

Programs

Magma

[a: n in [0..400] | IsPrime(a) where a is 2*n^2+1];// Vincenzo Librandi, Dec 02 2011

Mathematica

Select[Table[2n^2+1,{n,0,900}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

PARI

is(n)=isprime(2*n^2+1) \\ Charles R Greathouse IV, Jan 05 2013

Formula

a(n)=2*A089001(n)^2+1 = A000040(A090612(n)).

Extensions

Extended by Ray Chandler, Dec 21 2003