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

%I A057683 #44 Feb 17 2022 00:42:51
%S A057683 1,2,5,6,12,69,77,131,162,426,701,792,1221,1494,1644,1665,2129,2429,
%T A057683 2696,3459,3557,3771,4350,4367,5250,5670,6627,7059,7514,7929,8064,
%U A057683 9177,9689,10307,10431,11424,13296,13299,13545,14154,14286,14306,15137
%N A057683 Numbers k such that k^2 + k + 1, k^3 + k + 1 and k^4 + k + 1 are all prime.
%C A057683 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
%C A057683 For n > 1, no terms == 1 (mod 3) or == 3 (mod 5). - _Robert Israel_, Jul 31 2016
%H A057683 Reinhard Zumkeller and Robert Israel, <a href="/A057683/b057683.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..100 from Reinhard Zumkeller).
%e A057683 5 is included because 5^2 + 5 + 1 = 31, 5^3 + 5 + 1 = 131 and 5^4 + 5 + 1 = 631 are all prime.
%p A057683 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
%t A057683 Select[Range[16000],And@@PrimeQ/@(Table[n^i+n+1,{i,2,4}]/.n->#)&]  (* _Harvey P. Dale_, Mar 28 2011 *)
%o A057683 (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
%o A057683 (Haskell)
%o A057683 a057683 n = a057683_list !! (n-1)
%o A057683 a057683_list = filter (all (== 1) . p) [1..] where
%o A057683    p x = map (a010051 . (+ (x + 1)) . (x ^)) [2..4]
%o A057683 -- _Reinhard Zumkeller_, Nov 12 2012
%o A057683 (Python)
%o A057683 from sympy import isprime
%o A057683 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
%Y A057683 Cf. A049407.
%Y A057683 Cf. Subsequence of A219117; A010051.
%K A057683 easy,nice,nonn
%O A057683 1,2
%A A057683 _Harvey P. Dale_, Oct 20 2000