A128447 -id:A128447 - cp's OEIS frontend

A128448 Numbers k such that 8^k - k^8 is prime.

1, 11, 89, 201, 977, 1351, 3869, 60681

Offset: 1

Alexander Adamchuk, Mar 03 2007

From the Lifchitz link: 60681 and 85349 are also in this sequence. - Robert Price, Mar 27 2019

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.

lst={};k=8;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)

is(n)=isprime(8^n-n^8) \\ Charles R Greathouse IV, Feb 17 2017

a(6) and a(7) from Donovan Johnson, Feb 26 2008

a(8) confirmed (no intervening terms) by Michael S. Branicky, Jul 29 2024

A128449 Numbers k such that absolute value of 9^k - k^9 is prime.

Original entry on oeis.org

2, 10, 50, 7900, 18494, 23840, 36838

Offset: 1

Alexander Adamchuk, Mar 03 2007

Three larger terms 18494, 23840 and 36838 found by Donovan Johnson, Jul-Aug 2005.

From the Lifchitz link: 83024 is also a term. - Michael S. Branicky, Jul 29 2024

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128450, A128451, A122003, A128453, A128454.

lst={};k=9;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)

is(n)=isprime(abs(9^n-n^9)) \\ Charles R Greathouse IV, Feb 17 2017

a(4)-a(6) from Ryan Propper, Feb 22 2008

a(7) from Donovan Johnson, Feb 26 2008

A128450 Numbers k such that absolute value of 10^k - k^10 is prime.

Original entry on oeis.org

3, 9, 273, 399

Offset: 1

Alexander Adamchuk, Mar 03 2007

a(5) > 10^5 if it exists. - Michael S. Branicky, Nov 27 2024

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128451, A122003, A128453, A128454.

[n: n in [0..500]| IsPrime(10^n-n^10)]; // Vincenzo Librandi, Apr 01 2015

lst={};k=10;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0, 10000], PrimeQ[10^# - #^10] &] (* Vincenzo Librandi, Apr 01 2015 *)

is(n)=isprime(abs(10^n-n^10)) \\ Charles R Greathouse IV, Feb 17 2017

A128451 Numbers k such that the absolute value of 11^k - k^11 is prime.

Original entry on oeis.org

8, 14, 80, 212, 230, 1352, 13674, 16094, 44772

Offset: 1

Alexander Adamchuk, Mar 03 2007

Two larger terms 13674 and 16094 found by Donovan Johnson, Jul 2005.

Henri Lifchitz and Renaud Lifchitz, PRP Records.

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128450, A122003, A128453.

lst={};k=11;Do[If[PrimeQ[Abs[k^n-n^k]], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)

is(n)=ispseudoprime(abs(11^n-n^11)) \\ Charles R Greathouse IV, Feb 17 2017

a(6)-a(8) from Donovan Johnson, Feb 26 2008

a(9) discovered by Serge Batalov, entered by Robert Price, Apr 11 2019

A123206 Primes of the form x^y - y^x, for x,y > 1.

Original entry on oeis.org

7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, 9007199254738183, 79792265017612001, 1490116119372884249

Offset: 1

Alexander Adamchuk, Oct 04 2006

nonn

These are the primes in A045575, numbers of the form x^y - y^x, for x,y > 1. This includes all primes from A122735, smallest prime of the form (n^k - k^n) for k>1.

If y=1 was allowed, any prime p could be obtained for x=p+1. This motivates to consider sequence A243100 of primes of the form x^(y+1)-y^x. - M. F. Hasler, Aug 19 2014

			The primes 6102977801 and 1490116119372884249 are of the form 5^y-y^5 (for y=14 and y=26) and therefore members of this sequence. The next larger primes of this form would have y > 4500 and would be much too large to be included. - _M. F. Hasler_, Aug 19 2014

T. D. Noe, Table of n, a(n) for n=1..101 (terms < 10^400)
H. Lifchitz & R. Lifchitz, PRP of the form x^y-y^x on primenumbers.net.

Cf. A045575, A122735, A078202, A082754, A055651, A094133.
A163319 is the subsequences for fixed x=3, A243114 for x=6.
Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.

N:= 10^100: # to get all terms <= N
A:= NULL:
for x from 2 while x^(x+1) - (x+1)^x <= N do
   for y from x+1 do
      z:= x^y - y^x;
      if z > N then break
      elif z > 0 and isprime(z) then A:=A, z;
      fi
od od:
{A}; # Robert Israel, Aug 29 2014

Take[Select[Intersection[Flatten[Table[Abs[x^y-y^x],{x,2,120},{y,2,120}]]],PrimeQ[ # ]&],25]
nn=10^50; n=1; t=Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num0&&PrimeQ[#]&]],nn]] (* Harvey P. Dale, Nov 23 2013 *)

a=[];for(S=1,199,for(x=2,S-2,ispseudoprime(p=x^(y=S-x)-y^x)&&a=concat(a,p)));Set(a) \\ May be incomplete in the upper range of values, i.e., beyond a given S=x+y, a larger S may yield a smaller prime (for small x). - M. F. Hasler, Aug 19 2014

A128454 Numbers k such that the absolute value of 14^k - k^14 is prime.

Original entry on oeis.org

1, 5, 11, 89, 101, 579, 655, 8115

Offset: 1

Alexander Adamchuk, Mar 03 2007

a(9) > 50000. - Robert Price, Jun 17 2019

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128450, A128451, A122003, A128453.

Do[If[PrimeQ[Abs[14^n - n^14]], Print[n]], {n, 10^4}] (* Ryan Propper, Mar 27 2007 *)

is(n)=ispseudoprime(abs(14^n-n^14)) \\ Charles R Greathouse IV, Feb 17 2017

One more term from Ryan Propper, Mar 27 2007

A239279 Smallest k such that n^k - k^n is prime, or 0 if no such number exists.

Original entry on oeis.org

5, 1, 1, 14, 1, 20, 1, 10, 273, 14, 1, 38, 1, 68, 0

Offset: 2

Derek Orr, Mar 14 2014

It is believed that for all n > 4 and not in A097764, a(n) > 0.
a(n+1) = 1 if and only if n is prime.
If a(n) > 0 then a(n) and n are coprime.
If n is in the sequence A097764, then a(n) = 0 or 1 since n^k-k^n is factorable.
33^2570 - 2570^33 is a probable prime, so a(33) is probably 2570. - Jon E. Schoenfield, Mar 20 2014
Unknown a(n) values checked for k <= 10000 using PFGW. a(97) = 6006 found by Donovan Johnson in 2005. The Lifchitz link shows some large candidates for larger n but a smaller k exists in many of those cases. - Jens Kruse Andersen, Aug 13 2014
Unknown a(n) values checked for k <= 15000 using PFGW.

			2^1-1^2 = 1 is not prime. 2^2-2^2 = 0 is not prime. 2^3-3^2 = -1 is not prime. 2^4-4^2 = 0 is not prime. 2^5-5^2 = 7 is prime. So a(2) = 5.

H. Lifchitz and R. Lifchitz, PRP Top Records. Search for x^y-y^x
Derek Orr, Table of n, a(n) for n = 2..99 (unknown k-values are marked "unknown")

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128450.

a(n)=k=1; if(n>4, forprime(p=1, 100, if(ispower(n)&&ispower(n)%p==0&&n%p==0, return(0)); if(n%p==n, break))); k=1; while(!ispseudoprime(n^k-k^n), k++); return(k)
vector(15, n, a(n+1))

import sympy
from sympy import isprime
from sympy import gcd
def Min(x):
  k = 1
  while k < 5000:
    if gcd(k,x) == 1:
      if isprime(x**k-k**x):
        return k
      else:
        k += 1
    else:
      k += 1
x = 1
while x < 100:
  print(Min(x))
  x += 1

A243114 Primes of the form 6^x-x^6.

Original entry on oeis.org

5, 162287, 13055867207, 1719070799748422589190392551, 174588755932389037098918153698589675008087, 307180606913594390117978657628360735703373091543821695941623353827100004182413811352186951

Offset: 1

M. F. Hasler, Aug 20 2014

nonn

The next term is too large to include.
See A117706 for the corresponding numbers x.
The next term has 113 digits. - Harvey P. Dale, Jan 17 2018

Cf. A024068, A123206, A163319.

Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128450, A128451, A122003, A128453, A128454.

Select[Table[6^x-x^6,{x,200}],PrimeQ] (* Harvey P. Dale, Jan 17 2018 *)

for(x=1,1e5,ispseudoprime(p=6^x-x^6)&&print1(p","))

A173640 Primes of form n+2^n+3^n.

Original entry on oeis.org

2, 101, 60083, 11610630703530923996233764322611619865107483053157900065365853867349888133476404509

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 23 2010

nonn

For a(5), n > 10000. - Daniel Starodubtsev, Aug 04 2019

Cf. A082101, A082182, A128447, A128448, A128449, A128450, A128451, A100840, A100841

Select[Table[n+2^n+3^n,{n,0,6!}],PrimeQ[#]&]

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A128448 Numbers k such that 8^k - k^8 is prime.

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A128449 Numbers k such that absolute value of 9^k - k^9 is prime.

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A128450 Numbers k such that absolute value of 10^k - k^10 is prime.

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A128451 Numbers k such that the absolute value of 11^k - k^11 is prime.

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A123206 Primes of the form x^y - y^x, for x,y > 1.

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Maple

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A128454 Numbers k such that the absolute value of 14^k - k^14 is prime.

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A239279 Smallest k such that n^k - k^n is prime, or 0 if no such number exists.

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A243114 Primes of the form 6^x-x^6.

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