login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A123206 Primes of the form x^y - y^x, for x,y > 1. 6
7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, 9007199254738183, 79792265017612001, 1490116119372884249 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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]; num<nn, Sow[num]; While[k++; num=n^k-k^n; num<nn, Sow[num]]]][[2, 1]]]; Select[t, PrimeQ]

With[{nn=30}, Take[Sort[Select[#[[1]]^#[[2]]-#[[2]]^#[[1]]&/@Subsets[ Range[ 2nn], {2}], #>0&&PrimeQ[#]&]], nn]] (* Harvey P. Dale, Nov 23 2013 *)

PROG

(PARI) 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

CROSSREFS

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.

Sequence in context: A107693 A217717 A122528 * A035078 A177123 A124165

Adjacent sequences:  A123203 A123204 A123205 * A123207 A123208 A123209

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Oct 04 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 12 01:57 EST 2019. Contains 329948 sequences. (Running on oeis4.)