A057683 Numbers k such that k^2 + k + 1, k^3 + k + 1 and k^4 + k + 1 are all prime.

1, 2, 5, 6, 12, 69, 77, 131, 162, 426, 701, 792, 1221, 1494, 1644, 1665, 2129, 2429, 2696, 3459, 3557, 3771, 4350, 4367, 5250, 5670, 6627, 7059, 7514, 7929, 8064, 9177, 9689, 10307, 10431, 11424, 13296, 13299, 13545, 14154, 14286, 14306, 15137

Offset

1,2

Comments

After a(0) = 1, k^5 + k + 1 is never prime. Proof: k^5 + k + 1 = (k^2 + k + 1)*(k^3 - k^2 + 1). - Jonathan Vos Post, Oct 17 2007, edited by Robert Israel, Aug 01 2016

For n > 1, no terms == 1 (mod 3) or == 3 (mod 5). - Robert Israel, Jul 31 2016

Links

Reinhard Zumkeller and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..100 from Reinhard Zumkeller).

Example

5 is included because 5^2 + 5 + 1 = 31, 5^3 + 5 + 1 = 131 and 5^4 + 5 + 1 = 631 are all prime.

Maple

select(n -> isprime(n^4+n+1) and isprime(n^3+n+1) and isprime(n^2+n+1), [$1..50000]); # Robert Israel, Jul 31 2016

Mathematica

Select[Range[16000], And@@PrimeQ/@(Table[n^i+n+1, {i, 2, 4}]/.n->#)&]  (* Harvey P. Dale, Mar 28 2011 *)

Prog

(Magma) [n: n in [0..20000]|IsPrime(n^2+n+1) and IsPrime(n^3+n+1) and IsPrime(n^4+n+1)] // Vincenzo Librandi, Dec 20 2010
(Haskell)
a057683 n = a057683_list !! (n-1)
a057683_list = filter (all (== 1) . p) [1..] where
   p x = map (a010051 . (+ (x + 1)) . (x ^)) [2..4]
-- Reinhard Zumkeller, Nov 12 2012
(Python)
from sympy import isprime
A057683_list = [n for n in range(10**5) if isprime(n**2+n+1) and isprime(n**3+n+1) and isprime(n**4+n+1)] # Chai Wah Wu, Apr 02 2021

Crossrefs

Cf. A049407.

Cf. Subsequence of A219117; A010051.

Sequence in context: A257805 A082552 A243798 * A277012 A277022 A232603

Adjacent sequences: A057680 A057681 A057682 * A057684 A057685 A057686

Keyword

easy,nice,nonn

Author

Harvey P. Dale, Oct 20 2000

Status

approved