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At these values of k, the 1st, 2nd, 3rd and 4th cyclotomic polynomials all give prime numbers.
5

%I #23 Sep 24 2024 09:22:29

%S 6,150,2730,9000,9240,35280,41760,43050,53280,65520,76650,96180,

%T 111030,148200,197370,207480,213360,226380,254280,264600,309480,

%U 332160,342450,352740,375450,381990,440550,458790,501030,527070,552030,642360,660810

%N At these values of k, the 1st, 2nd, 3rd and 4th cyclotomic polynomials all give prime numbers.

%C Numbers k such that k-1, k+1, k^2+k+1 and k^2+1 are all primes.

%H Amiram Eldar, <a href="/A070025/b070025.txt">Table of n, a(n) for n = 1..10000</a>

%e For k = 6: 5, 7, 43 and 37 are prime values of the first 4 cyclotomic polynomials.

%t lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[1+n+n^2]&&PrimeQ[1+n^2], AppendTo[lst, n]], {n, 10^6}];lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 19 2008 *)

%t Select[Range[10^6], Function[k, AllTrue[Cyclotomic[#, k] & /@ Range@ 4, PrimeQ]]] (* _Michael De Vlieger_, Jul 18 2017 *)

%o (PARI) is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^2+k+1); \\ _Amiram Eldar_, Sep 24 2024

%Y Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070020, A070042.

%K easy,nonn

%O 1,1

%A _Labos Elemer_, May 07 2002