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 1-10 of 12 results. Next

A110932 Numbers k such that 2*k^k + 1 is prime.

Original entry on oeis.org

0, 1, 12, 18, 251, 82992
Offset: 0

Views

Author

Ray G. Opao, Sep 25 2005

Keywords

Comments

As a "list of numbers such that ...", the sequence should have offset 1, but to preserve the validity of formulas referring to this sequence, the offset was set to 0 when the initial value a(0)=0 was added. - M. F. Hasler, Sep 02 2012

Crossrefs

Cf. A110931, A121270 (= primes in A014566), A088790, A160360, A160600.
The primes 2n^n+1, for k<4, n=a(k)<251, are listed at A216148(k) = A216147(a(k)). - M. F. Hasler, Sep 02 2012

Programs

  • Mathematica
    Join[{0}, Select[Range[1000], PrimeQ[2*#^# + 1] &]] (* Robert Price, Mar 27 2019 *)
  • PARI
    is_A110932(n)=ispseudoprime(n^n*2+1) \\ M. F. Hasler, Sep 02 2012

Extensions

a(5) from Serge Batalov, Apr 08 2018

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.

A301808 Primes of form 4*k^k + 3.

Original entry on oeis.org

7, 19, 12503, 40000000003, 8203095294242440214580822436689504141944719346082430189179667864757469187
Offset: 1

Views

Author

Seiichi Manyama, Mar 27 2018

Keywords

Comments

The next term is too large to include.

Crossrefs

Primes of form b*k^k + b - 1: A216148 (b=2), this sequence (b=4).
Cf. A301520, A301641.

Formula

a(n) = 4*A301520(n+1)^A301520(n+1) + 3.

A160600 Numbers k such that 3*(2k)^(2k)+1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 143, 225
Offset: 1

Views

Author

M. F. Hasler, Jul 10 2009

Keywords

Comments

This corresponds to the numbers such that 3m^m+1 is prime, but these must all be even, m=2k, and therefore it is more natural to record the sequence of k=m/2.
Next term > 15000. - Matevz Markovic, Oct 09 2012

Examples

			a(1) = 1, because 2^2*3+1 = 13 is the smallest prime of this form.
a(2) = 2, because 4^4*3+1 = 769 is the next smallest prime of this form. a(3) = 3, because 6^6*3+1 = 139969 is again a prime.

Crossrefs

Cf. A160360 (3n^n+2 is prime), A121270 = primes among Sierpinski numbers A014566(n)=n^n+1; A216148 = A216147(A110932): primes 2n^n+1; A088790, A065798.

Programs

  • Maple
    q:= k-> isprime(3*(2*k)^(2*k)+1):
select(q, [$1..225])[];  # Alois P. Heinz, Aug 04 2025
  • PARI
    for(i=1,9999,ispseudoprime(i^i*3+1)&print1(i/2,","))

A216147 2*n^n + 1.

Original entry on oeis.org

3, 3, 9, 55, 513, 6251, 93313, 1647087, 33554433, 774840979, 20000000001, 570623341223, 17832200896513, 605750213184507, 22224013651116033, 875787780761718751, 36893488147419103233, 1654480523772673528355, 78692816150593075150849, 3956839311320627178247959
Offset: 0

Views

Author

M. F. Hasler, Sep 02 2012

Keywords

Crossrefs

See A216148 for primes in this sequence, A110932 for the corresponding indices.

A301642 Primes of form 16*k^k + 1.

Original entry on oeis.org

17, 433, 746497, 142657607172097
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 108 digits; a(6) has 166 digits; a(7) has 170 digits. - Harvey P. Dale, Aug 23 2019

Crossrefs

Primes of form b*k^k + 1: A121270 (b=1), A216148 (b=2), A301644 (b=3), A301641 (b=4), this sequence (b=16).
Cf. A301522.

Programs

  • Mathematica
    Select[Table[16*k^k+1,{k,20}],PrimeQ] (* Harvey P. Dale, Aug 23 2019 *)

Formula

a(n) = 16*A301522(n+1)^A301522(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.

A302091 Primes of form 6*k^k + 5.

Original entry on oeis.org

11, 29, 167, 279941, 4941263
Offset: 1

Views

Author

Seiichi Manyama, Apr 01 2018

Keywords

Comments

The next term is too large to include.
The next term has 286 digits. - Harvey P. Dale, Oct 24 2021

Crossrefs

Primes of form b*k^k + b - 1: A216148 (b=2), A301811 (b=3), A301808 (b=4), A302089 (b=5), this sequence (b=6).
Cf. A302090.

Programs

  • Mathematica
    Select[Table[6 n^n + 5, {n, 20}], PrimeQ] (* Harvey P. Dale, Oct 24 2021 *)

Formula

a(n) = 6*A302090(n+1)^A302090(n+1) + 5.

A301811 Primes of form 3*k^k + 2.

Original entry on oeis.org

5, 83, 9377, 2470631
Offset: 1

Views

Author

Seiichi Manyama, Mar 27 2018

Keywords

Comments

The next term is too large to include.
This sequence is different from A216146.

Crossrefs

Primes of form b*k^k + b - 1: A216148 (b=2), this sequence (b=3), A301808 (b=4).
Cf. A160360, A301644.

Formula

a(n) = 3*A160360(n+1)^A160360(n+1) + 2.
Showing 1-10 of 12 results. Next

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