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A057683 Numbers n such that n^2+n+1, n^3+n+1 and n^4+n+1 are all prime. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

After a(0) = 1, it is never the case that n^5 + n + 1 is prime. Proof: n^5+n+1 = (n^2+n+1)*(n^3-n^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

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

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Last modified December 13 08:43 EST 2018. Contains 318082 sequences. (Running on oeis4.)