A000059 Numbers k such that (2k)^4 + 1 is prime.

1, 2, 3, 8, 10, 12, 14, 17, 23, 24, 27, 28, 37, 40, 41, 44, 45, 53, 59, 66, 70, 71, 77, 80, 82, 87, 90, 97, 99, 102, 105, 110, 114, 119, 121, 124, 127, 133, 136, 138, 139, 144, 148, 156, 160, 164, 167, 170, 176, 182, 187, 207, 215, 218, 221, 233, 236, 238, 244, 246

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			(2 * 2)^4 + 1 = 4^4 + 1 = 17, which is prime, so 2 is in the sequence.
(2 * 3)^4 + 1 = 6^4 + 1 = 1297, which is prime, so 3 is in the sequence.
(2 * 4)^4 + 1 = 8^4 + 1 = 4097 = 17 * 241, so 4 is not in the sequence.
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• J. Bohman, New primes of the form n^4+1, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 370-372.
• M. Lal, Primes of the form n^4 + 1, Math. Comp., 21 (1967), 245-247.

Crossrefs

Cf. A037896 (primes of the form n^4 + 1).

Programs

Magma

[n: n in [1..10000] | IsPrime((2*n)^4+1)] # Vincenzo Librandi, Nov 18 2010

Maple

A000059:=n->`if`(isprime((2*n)^4+1),n,NULL): seq(A000059(n), n=1..250); # Wesley Ivan Hurt, Aug 26 2014

Mathematica

Select[Range[300], PrimeQ[(2 * #)^4 + 1] &] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

PARI

for(n=1,10^3,if(isprime( (2*n)^4+1 ),print1(n,", "))) \\ Hauke Worpel (thebigh(AT)outgun.com), Jun 11 2008 [edited by Michel Marcus, Aug 27 2014]

Python

from sympy import isprime
print([n for n in range(10**3) if isprime(16*n**4+1)])
# Derek Orr, Aug 27 2014

Formula

a(n) = A000068(n+1)/2 for n >= 1. [Corrected by Jianing Song, Feb 03 2019]

Extensions

More terms from Hugo Pfoertner, Aug 27 2003