A070025 At these values of k, the 1st, 2nd, 3rd and 4th cyclotomic polynomials all give prime numbers.

6, 150, 2730, 9000, 9240, 35280, 41760, 43050, 53280, 65520, 76650, 96180, 111030, 148200, 197370, 207480, 213360, 226380, 254280, 264600, 309480, 332160, 342450, 352740, 375450, 381990, 440550, 458790, 501030, 527070, 552030, 642360, 660810

Offset: 1

Author

Labos Elemer, May 07 2002

Keywords

• easy
• nonn

Comments

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

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

PARI

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^2+k+1); \\ Amiram Eldar, Sep 24 2024