cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Labos Elemer, May 07 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Labos Elemer, May 07 2002

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070020, A070025.

Programs

  • PARI
    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

Extensions

More terms from Don Reble, May 11 2002
a(24)-a(28) from Amiram Eldar, Sep 24 2024

A070024 First prime > 10^n in which every substring of length n is prime.

Original entry on oeis.org

23, 113, 1013, 10139, 100379, 1000037, 10000379, 100000193, 1000001237, 10000000097, 100000000193, 1000000000193, 10000000001777, 100000000001831, 1000000000036931, 10000000000001873, 100000000000000691
Offset: 1

Views

Author

Robert G. Wilson v, Apr 12 2002

Keywords

Crossrefs

Cf. A019546, A069488, A069489, A070025.
Cf. A179335. [From Reinhard Zumkeller, Jul 11 2010]

Programs

  • Mathematica
    Do[k = 10^n; While[ !PrimeQ[k] || Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[k], n, 1]]]] != {True}, k++ ]; Print[k], {n, 1, 25}]

A070737 Smallest number x such that the first n cyclotomic polynomials evaluated at x are primes.

Original entry on oeis.org

3, 4, 6, 6, 1068630, 6770610, 2981997480, 339126523890, 120351747887280, 13533264289711320
Offset: 1

Views

Author

Don Reble, May 15 2002 and Phil Carmody, Aug 09 2002

Keywords

Comments

Let Phi_k be the k-th cyclotomic polynomial. a(n) is the least integer x such that for each k from 1 to n, Phi_k(x) is prime.

Examples

			a(6)=x=6770610 because Phi_1(x)=x-1, Phi_2(x)=x+1, Phi_3(x)=x^2+x+1, Phi_4(x)=x^2+1, Phi_5(x)=x^4+x^3+x^2+x+1 and Phi_6(x)=x^2-x+1 are all prime.

Crossrefs

Cf. A014574, A070020, A070025, A070043.
Showing 1-4 of 4 results.

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