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 1-5 of 5 results.

A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

Original entry on oeis.org

4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840
Offset: 1

Views

Author

Labos Elemer, Apr 23 2002

Keywords

Comments

Essentially the same as A129293. - R. J. Mathar, Jun 14 2008
Solutions to the equation: A000005(n^4-1) = 8. - Enrique Pérez Herrero, May 03 2012
Terms > 6 are multiples of 30. Subsequence of A070689. - Zak Seidov, Nov 12 2012
{a(n)-1} is a subsequence of A157468; for n>1, {a(n)^2+2} is a subsequence of A242720. - Vladimir Shevelev, Aug 31 2014

Examples

			150 is a term since 149, 151 and 22501 are all primes.

Links

Crossrefs

Cf. A000005, A000720, A001359, A006512, A014574, A002496, A005574, A070156, A070689, A129293, A157468, A242720.

Programs

  • Maple
    select(n -> isprime(n-1) and isprime(n+1) and isprime(n^2+1), [seq(2*i,i=1..10000)]); # Robert Israel, Sep 02 2014
  • Mathematica
    Do[s=n; If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[1+s^2], Print[n]], {n, 1, 1000000}]
Select[Range[22000],AllTrue[{#+1,#-1,#^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 19 2014 *)
  • PARI
    is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1); \\ Amiram Eldar, Apr 15 2024

Formula

For n>1, a(n)^2 = A242720(pi(a(n)-2)) - 2, where pi(n) is the prime counting function (A000720). - Vladimir Shevelev, Sep 02 2014

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, May 07 2002

Keywords

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

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

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

A070157 Numbers k such that k-1, k+1, k^2+1, k^4+1 and k^8+1 are all prime numbers.

Original entry on oeis.org

4, 19380, 9443670, 11054760, 15992070, 22482330, 32557380, 51102510, 57978840, 60549240, 64671570, 84045960, 89757960, 111316170, 112821690, 116433510, 171124380, 171418650, 183082350, 196694760, 197021160, 241803240, 266498460
Offset: 1

Views

Author

Labos Elemer, Apr 23 2002

Keywords

Examples

			19380 is a term since 19380-1 = 19379, 19380+1 = 19381, 19380^2+1 = 375584401, 19380^4+1 = 141063641523360001 and 19380^8+1 = 19898950959831015581425689600000001 are primes.

Links

Crossrefs

Subsequence of A070155 and A070156.

Programs

  • Mathematica
    Do[p = Prime[n] + 1; If[ PrimeQ[p + 1] && PrimeQ[1 + p^2] && PrimeQ[1 + p^4] && PrimeQ[1 + p^8], Print[p]], {n, 1, 115000000}]
Select[Range[2665*10^5],AllTrue[{#-1,#+1,#^2+1,#^4+1,#^8+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 03 2019 *)
  • PARI
    is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^4+1) && isprime(k^8+1); \\ Amiram Eldar, Jun 26 2024

Extensions

Edited and extended by Robert G. Wilson v, May 04 2002

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
Showing 1-5 of 5 results.

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