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 1-8 of 8 results.

A100407 Numbers k such that k^k+2 is prime.

Original entry on oeis.org

1, 3, 737, 1349
Offset: 1

Views

Author

Farideh Firoozbakht, Nov 19 2004

Keywords

Comments

All terms are odd. Next term is greater than 3500. 1349^1349+2 is a probable prime with length 4223.
The next term, if it exists, is greater than 50000. - Hugo Pfoertner, Jun 28 2019
The next term, if it exists, is greater than 430000. - Ryan Propper, Dec 15 2023

Examples

			737 is in the sequence because 737^737+2 is prime.

Crossrefs

Cf. A100408.

Programs

  • Mathematica
    Do[If[PrimeQ[n^n+2], Print[n]], {n, 3500}]
  • PARI
    is(n)=ispseudoprime(n^n+2) \\ Charles R Greathouse IV, May 22 2017

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

A162927 Primes of the form n^n-2.

Original entry on oeis.org

2, 823541, 1978419655660313589123977, 5842587018385982521381124419, 88817841970012523233890533447265623, 66009724686219550843768321818371771650147004059278069406814190436565131829325062447
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 17 2009

Keywords

Crossrefs

Main sequence is A100408.

Programs

  • Mathematica
    f[n_]:=n^n-2; lst={};Do[If[PrimeQ[f[n]],AppendTo[lst,f[n]]],{n,5!}];lst
Join[{2},Select[Table[n^n-2,{n,1,51,2}],PrimeQ]] (* Harvey P. Dale, Feb 12 2023 *)

Extensions

Cross-reference from Charles R Greathouse IV, Oct 08 2009

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.

A300292 Numbers k such that k^k + 9 is a prime.

Original entry on oeis.org

2, 130, 140
Offset: 1

Views

Author

Seiichi Manyama, Mar 16 2018

Keywords

Comments

No more terms <= 7000. - Jon E. Schoenfield, Mar 16 2018
No more terms <= 30000. - Michael S. Branicky, Sep 02 2024

Crossrefs

Numbers k such that k^k + b is a prime: A300981 (b=-10), A300976 (b=-5), A100408 (b=-2), A100407 (b=2), A166852 (b=3), A100837 (b=4), A100838 (b=7), this sequence (b=9), A100839 (b=10), A173974 (b=43).
Cf. A074966.

Programs

  • PARI
    isok(k) = ispseudoprime(k^k + 9); \\ Altug Alkan, Mar 16 2018

A300976 Numbers k such that k^k - 5 is a prime.

Original entry on oeis.org

4, 104, 124, 728
Offset: 1

Views

Author

Seiichi Manyama, Mar 17 2018

Keywords

Comments

728^728 - 5 is a probable prime.
Next term, if it exists, is greater than 5000. - Vaclav Kotesovec, Mar 25 2018
Next term, if it exists, is greater than 31000. - Robert Price, Mar 26 2018

Crossrefs

Numbers k such that k^k + b is a prime: A300981 (b=-10), this sequence (b=-5), A100408 (b=-2), A100407 (b=2), A166852 (b=3), A100837 (b=4), A100838 (b=7), A300292 (b=9), A100839 (b=10), A173974 (b=43).

Programs

  • Mathematica
    Select[Range[1000], PrimeQ[#^# - 5] &] (* Vaclav Kotesovec, Mar 25 2018 *)
  • PARI
    isok(k) = ispseudoprime(k^k - 5); \\ Altug Alkan, Mar 17 2018

A300981 Numbers k such that k^k - 10 is a prime.

Original entry on oeis.org

3, 9, 27, 249
Offset: 1

Views

Author

Seiichi Manyama, Mar 17 2018

Keywords

Comments

a(5), if it exists, is greater than 5000. - Vaclav Kotesovec, Mar 25 2018
a(5), if it exists, is greater than 25000. - Michael S. Branicky, Sep 02 2024

Crossrefs

Numbers k such that k^k + b is a prime: this sequence (b=-10), A300976 (b=-5), A100408 (b=-2), A100407 (b=2), A166852 (b=3), A100837 (b=4), A100838 (b=7), A300292 (b=9), A100839 (b=10), A173974 (b=43).

Programs

  • Mathematica
    Select[Range[1000], PrimeQ[#^# - 10] &] (* Vaclav Kotesovec, Mar 25 2018 *)
  • PARI
    isok(k) = ispseudoprime(k^k - 10); \\ Altug Alkan, Mar 17 2018

A348329 Numbers k such that k' = k'', where ' is the arithmetic derivative.

Original entry on oeis.org

0, 1, 4, 27, 3125, 823543, 1647082, 2238771
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 12 2021

Keywords

Comments

For n > 2, a(n) is such that a(n)' = p^p for some prime p. So A051674 is a subsequence. - David A. Corneth, Oct 13 2021
If p > 2 and p^p-2 are both primes (i.e., p is an odd prime term of A100408), then 2*(p^p-2) is a term. Terms of this type are 1647082 and 3956839311320627178247954, corresponding to p = 7 and 19 respectively. - Amiram Eldar, Oct 13 2021

Crossrefs

Cf. A003415, A051674, A068346, A100408.

Programs

  • Maple
    isA348329 := proc(n)
    local d ;
    d := A003415(n) ;
    if A003415(d) = d then
        true ;
    else
        false;
    end if;
end proc:
for n from 0 do
    if isA348329(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Oct 19 2021
  • Mathematica
    d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 2.5*10^6], d[#] == d[d[#]] &] (* Amiram Eldar, Oct 13 2021 *)
  • PARI
    ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = ad(k) == ad(ad(k)); \\ Michel Marcus, Oct 18 2021
  • Python
    from sympy import factorint
from itertools import count, islice
def ad(n): return 0 if n<2 else sum(n*e//p for p, e in factorint(n).items())
def agen(): yield from (k for k in count(0) if (adk:=ad(k)) == ad(adk))
print(list(islice(agen(), 5))) # Michael S. Branicky, Oct 12 2022

Formula

Numbers k such that A003415(k) = A068346(k).
Showing 1-8 of 8 results.

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