A000059 Numbers k such that (2k)^4 + 1 is prime. (Formerly M0867 N0332)

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

1,2

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.

Formula

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

Example

(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.

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 *)

Prog

(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]
(Magma)[n: n in [1..10000] | IsPrime((2*n)^4+1)] # Vincenzo Librandi, Nov 18 2010
(Python)
from sympy import isprime
print([n for n in range(10**3) if isprime(16*n**4+1)])
# Derek Orr, Aug 27 2014

Crossrefs

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

Sequence in context: A325424 A057543 A190650 * A340301 A216761 A276559

Adjacent sequences: A000056 A000057 A000058 * A000060 A000061 A000062

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Hugo Pfoertner, Aug 27 2003

Status

approved