A096170 Primes of the form (k^4 + 1)/2.

41, 313, 1201, 7321, 14281, 41761, 97241, 139921, 353641, 750313, 1156721, 5278001, 6922921, 8925313, 12705841, 14199121, 21523361, 56275441, 60775313, 81523681, 87450313, 100266961, 138461441, 273990641, 370600313, 407865361

Offset: 1

Author

Hugo Pfoertner, Jun 19 2004

Keywords

• nonn

Comments

Note that k must be odd. Terms of primitive Pythagorean triples: (k^2, (k^4-1)/2, (k^4+1)/2).

Examples

			a(1)=41 because (3^4 + 1)/2 = 82/2 = 41 is prime.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A096169 (n^4+1)/2 is prime, A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.

Programs

Magma

[ a: n in [0..2500] | IsPrime(a) where a is ((n^4+1) div 2) ]; // Vincenzo Librandi, Apr 15 2011

Mathematica

Select[(Range[200]^4+1)/2,PrimeQ] (* Harvey P. Dale, Mar 09 2013 *)

PARI

list(lim)=my(v=List(),t); forstep(n=3,sqrtnint(lim\1*2-1,4),2, if(isprime(t=(n^4+1)/2), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

Extensions

Name edited by Zak Seidov, Apr 14 2011