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.

A166852 Numbers k such that k^k + 3 is prime.

Original entry on oeis.org

2, 1036, 2770
Offset: 1

Views

Author

Farideh Firoozbakht, Nov 20 2009

Keywords

Comments

Numbers corresponding to a(2) and a(3) are probable primes. 2770 is in the sequence so 2770^2770 + 3 is a probable prime; it is interesting that 277027703 is also prime. For the first term we have the same property: both 2^2 + 3 and 223 are prime.
For k = -1, k^k + 3 = 2 is prime but sequence focuses on the positive values of k. - Altug Alkan, Nov 28 2015
a(4) > 25000. - Michael S. Branicky, Oct 15 2024

Crossrefs

Cf. A087037, A100407, A100408, A166853.

Programs

  • Mathematica
    Do[If[GCD[n,3]==1&&PrimeQ[n^n+3],Print[n]],{n,2,5362,2}]
  • PARI
    is(n)=ispseudoprime(n^n+3) \\ Charles R Greathouse IV, Jun 13 2017

A087038 Smallest integer x > 1 such that x^x + n is prime, or 0 if no such x exists.

Original entry on oeis.org

2, 3, 2, 3, 444
Offset: 1

Views

Author

Hugo Pfoertner, Jul 31 2003

Keywords

Comments

The sequence with the unknown terms indicated by ?: 2, 3, 2, 3, 444, ?, 2, ?, 2, 3, ?, 5, 2, 3, 2, 3, ?, 19, 2, 3, 4, 19, 6, ?, 2, 3, 2, 15, 30, 7, 6, 3, 2, 3, 6, ?, 2, 5, 2, 3, ...
The unknown terms a(6), a(8), a(11), a(17), a(24), a(36) are > 6000.
It is conjectured that such x always exists. - Dean Hickerson
We can show that for all n=(6k-1)^3, k > 0, there is no such x, which disproves the conjecture. See the main entry A087037 for more details. - Farideh Firoozbakht and M. F. Hasler, Nov 27 2009

Examples

			a(4)=3 because 3^3 + 4 = 27 + 4 = 31 is prime.

Links

Crossrefs

Cf. A000312 (n^n), A087037 (x^x+n is prime, x>0).

Extensions

Name edited by Altug Alkan, Apr 01 2018

A166853 a(n) is the smallest number m such that m^m-n is prime, or zero if there is no such m.

Original entry on oeis.org

2, 2, 8, 3, 4, 5, 6, 3, 0, 3, 78, 13, 6, 3, 4, 3, 4, 17, 12, 3, 118, 3, 4, 3, 3
Offset: 1

Views

Author

Farideh Firoozbakht and M. F. Hasler, Nov 27 2009

Keywords

Comments

The sequence with the unknown terms a(n) indicated by -n:
(0's occur for n=9, 49, 81, 121....)
2,2,8,3,4,5,6,3,0,3,78,13,6,3,4,3,4,17,12,3,118,3,4,3,3,
-26,4,-28,4,487,90,9,4,-34,24,5,6,271,28,969,-41,5,-43,7,4,5,32,37,0,621,
20,15,34,7,6,9,4,5,4,7,-61,7,4,5,4,-66,6,63,134,27,10,35,102,31,4,
5,4,569,-79,13,0,15,4,5,-85,7,110,5,4,131,1122,7,4,11,8,7,6,9,4,-100,
22,5,-103,-104,4,5,4,11,12,39,-111,...
If they exist, the first two unknown terms, a(26) and a(28), they are greater than 10000. All other unknown terms a(n), for n<112 are greater than 4000.
If it exists, a(26) > 25000. - Robert Price, Apr 26 2019

Examples

			We have a(1)=2 since 1^1-1 is not prime, but 2^2-1 is prime.
a(9)=0 since 2^2-9 is not prime, and if m is an even number greater than 2 then m^m-9=(m^(m/2)-3)*(m^(m/2)+3) is composite. So there is no number m such that m^m-9 is prime. The same applies to any odd square > 25.
We have a(25)=3 since 3^3-25=2 is prime. But 25 is the only known square of the form m^m-2, so a(n)=0 for other odd squares > 25, e.g., n = 49,81,121,....
a(115)=2736 is the largest known term. 2736^2736-115 is a probable prime.

Crossrefs

Cf. A087037, A087038, A016754, A016923, A100407, A100408, A166852.

Formula

a(n)=0 if n=3^2 or n=(2k+1)^2 > 25, or n = (6k+1)^3 = A016923(k) with k>0.

A118710 Smallest positive integer k such that k^k + F(n) is prime, where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 2, 444
Offset: 1

Views

Author

Ryan Propper, May 20 2006

Keywords

Comments

Next term is not known. Sequence continues: 1, 1, 1, 2, 444, ?, 2, 4, 3, 2, ?, ?, 6, ?, 1059, 2, 2, ?, ?, 14, 3, 66, 2, ?, 2, 46, 15, 8, 78, 273, 2, 2. All unknown terms are >= 2000. All known terms except a(15) = 1059 correspond to certified primes.
a(6) = A087037(8) > 30300.

Programs

  • Mathematica
    Do[k = 1; While[ !PrimeQ[k^k + Fibonacci[n]], k++ ]; Print[k], {n, 32}]

Formula

a(n) = A087037(A000045(n)).
Showing 1-4 of 4 results.

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