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.

A121270 Prime Sierpinski numbers of the first kind: primes of the form k^k+1.

Original entry on oeis.org

2, 5, 257
Offset: 1

Views

Author

Alexander Adamchuk, Aug 23 2006

Keywords

Comments

Sierpinski proved that k>1 must be of the form 2^(2^j) for k^k+1 to be a prime. All a(n) > 2 must be the Fermat numbers F(m) with m = j+2^j = A006127(j). [Edited by Jeppe Stig Nielsen, Jul 09 2023]

References

  • See e.g. pp. 156-157 in M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001. - Walter Nissen, Mar 20 2010

Links

Crossrefs

Primes of form b*k^k + 1: this sequence (b=1), A216148 (b=2), A301644 (b=3), A301641 (b=4), A301642 (b=16).
Cf. A014566, A048861, A006127, A000215.

Programs

  • Mathematica
    Do[f=n^n+1;If[PrimeQ[f],Print[{n,f}]],{n,1,1000}]
  • PARI
    for(n=1,9,if(ispseudoprime(t=n^n+1),print1(t", "))) \\ Charles R Greathouse IV, Feb 01 2013

Extensions

Definition rewritten by Walter Nissen, Mar 20 2010

A301641 Primes of form 4*k^k + 1.

Original entry on oeis.org

5, 17, 109, 3294173, 355271367880050092935562133789062501
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2018

Keywords

Comments

No additional terms through k=1000. - Harvey P. Dale, Oct 06 2023

Crossrefs

Primes of form b*k^k + 1: A121270 (b=1), A216148 (b=2), A301644 (b=3), this sequence (b=4), A301642 (b=16).
Cf. A301519, A301808.

Programs

  • Maple
    a:=k->`if`(isprime(4*k^k+1),4*k^k+1,NULL): seq(a(k),k=1..1400); # Muniru A Asiru, Mar 25 2018
  • Mathematica
    Select[Table[4n^n+1,{n,30}],PrimeQ] (* Harvey P. Dale, Oct 06 2023 *)

Formula

a(n) = 4*A301519(n+1)^A301519(n+1) + 1.

A301644 Primes of form 3*k^k + 1.

Original entry on oeis.org

13, 769, 139969, 30000000001
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2018

Keywords

Comments

The next term is too large to include.
The next term (a(5)) has 703 digits. - Harvey P. Dale, Sep 04 2018

Crossrefs

Primes of form b*k^k + 1: A121270 (b=1), A216148 (b=2), this sequence (b=3), A301641 (b=4), A301642 (b=16).
Cf. A160600.

Programs

  • Mathematica
    Select[Table[3*k^k+1,{k,500}],PrimeQ] (* Harvey P. Dale, Sep 04 2018 *)

Formula

a(n) = 3*(2*A160600(n))^(2*A160600(n)) + 1.

A301645 Primes of form 16*k^k - 1.

Original entry on oeis.org

431, 49999, 7006302246093749999, 7094823811888604319171994089906398284847
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2018

Keywords

Crossrefs

Cf. A301524, A301642.

Formula

a(n) = 16*A301524(n)^A301524(n) - 1.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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