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%I M0867 N0332 #60 Jan 21 2022 03:42:08
%S 1,2,3,8,10,12,14,17,23,24,27,28,37,40,41,44,45,53,59,66,70,71,77,80,
%T 82,87,90,97,99,102,105,110,114,119,121,124,127,133,136,138,139,144,
%U 148,156,160,164,167,170,176,182,187,207,215,218,221,233,236,238,244,246
%N Numbers k such that (2k)^4 + 1 is prime.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000059/b000059.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Bohman, <a href="https://doi.org/10.1007/BF01951947">New primes of the form n^4+1</a>, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 370-372.
%H M. Lal, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0222007-9">Primes of the form n^4 + 1</a>, Math. Comp., 21 (1967), 245-247.
%F a(n) = A000068(n+1)/2 for n >= 1. [Corrected by _Jianing Song_, Feb 03 2019]
%e (2 * 2)^4 + 1 = 4^4 + 1 = 17, which is prime, so 2 is in the sequence.
%e (2 * 3)^4 + 1 = 6^4 + 1 = 1297, which is prime, so 3 is in the sequence.
%e (2 * 4)^4 + 1 = 8^4 + 1 = 4097 = 17 * 241, so 4 is not in the sequence.
%p A000059:=n->`if`(isprime((2*n)^4+1),n,NULL): seq(A000059(n), n=1..250); # _Wesley Ivan Hurt_, Aug 26 2014
%t Select[Range[300], PrimeQ[(2 * #)^4 + 1] &] (* _Vladimir Joseph Stephan Orlovsky_, Jan 24 2012 *)
%o (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]
%o (Magma)[n: n in [1..10000] | IsPrime((2*n)^4+1)] # _Vincenzo Librandi_, Nov 18 2010
%o (Python)
%o from sympy import isprime
%o print([n for n in range(10**3) if isprime(16*n**4+1)])
%o # _Derek Orr_, Aug 27 2014
%Y Cf. A037896 (primes of the form n^4 + 1).
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Hugo Pfoertner_, Aug 27 2003
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