A070042 -id:A070042 - cp's OEIS frontend

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

Labos Elemer, May 07 2002

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

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

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

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

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 *)

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

A070020 At these values of k the first, 2nd and 3rd cyclotomic polynomials all give prime numbers.

Original entry on oeis.org

6, 12, 138, 150, 192, 348, 642, 1020, 1092, 1230, 1620, 1788, 1932, 2112, 2142, 2238, 2658, 2688, 2730, 3330, 3540, 3918, 4002, 4158, 5010, 5640, 6090, 6450, 6552, 6702, 7950, 8088, 9000, 9042, 9240, 9462, 9768, 10008, 10092, 10272, 10302, 10332

Offset: 1

Labos Elemer, May 07 2002

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

			For k = 6: 5, 7 and 43 are prime values of the first 3 cyclotomic polynomials.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michael De Vlieger)

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070025, A070042.

psQ[n_]:=And@@PrimeQ[{n-1,n+1,n^2+n+1}]; Select[Range[11000],psQ] (* Harvey P. Dale, Nov 05 2011 *)
Select[Range[10500], AllTrue[Cyclotomic[Range@ 3, #], PrimeQ] &] (* Michael De Vlieger, Dec 08 2018 *)

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

cp's OEIS Frontend

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

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