A070157 -id:A070157 - 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

A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

Original entry on oeis.org

1068630, 1441590, 1867950, 3429300, 4084230, 5651730, 6322890, 6770610, 7158630, 7804830, 9437760, 9624270, 13625850, 23194860, 25848840, 26588520, 28714950, 29451840, 32984430, 33650580, 36500910, 38177130, 42856590, 49531020, 50016540, 50222070, 52083330, 54637590

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that C1(k) = k-1, C2(k) = k+1, C3(k) = k^2+k+1, C4(k) = k^2+1 and C5(k) = k^4+k^3+k^2+k+1 are all primes.

			For k = 1068630: the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials give a quintet of primes: {1068629, 1068631, 1141971145531, 1141970076901, 1304096876879617162402531}.

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, A070025.

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^2+k+1) && isprime(k^4+k^3+k^2+k+1) ; \\ Amiram Eldar, Sep 24 2024

More terms from Don Reble, May 11 2002

a(24)-a(28) from 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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A070020 At these values of k the first, 2nd and 3rd cyclotomic polynomials all give prime numbers.

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A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

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