A094133 Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1.

3, 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, 4318114567396436564035293097707729426477458833, 5052785737795758503064406447721934417290878968063369478337

Offset

1,1

Comments

Contains A061119 as a subsequence.

Links

Charles R Greathouse IV and Hans Havermann (Charles R Greathouse IV to 49), Table of n, a(n) for n = 1..100

Ed Copeland and Brady Haran, Leyland Numbers, Numberphile video (2014)

Hans Havermann, Table of n (where known), Leyland index, number of digits in decimal representation, and (x,y) pair for all known solutions

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Paul Leyland, Primes and PRPs of the form x^y + y^x

Example

2^1 + 1^2, 3^2 + 2^3, 9^2 + 2^9, 15^2 + 2^15, 21^2 + 2^21, 33^2 + 2^33, 24^5 + 5^24, 56^3 + 3^56, 32^15 + 15^32, 54^7 + 7^54, 38^33 + 33^38.

Maple

N:= 10^100: # to get all terms <= N
A:= {3}:
for n from 2 while 2*n^n < N do
  for k from n+1 do if igcd(n, k)=1 then
     a:= n^k + k^n;
     if a > N then break fi;
     if isprime(a) then A:= A union {a} fi fi;
  od
od:
A; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(A, list)); # Robert Israel, Apr 13 2015

Mathematica

a = {3}; Do[Do[k = m^n + n^m; If[PrimeQ[k], AppendTo[a, k]], {m, 2, n}], {n, 2, 100}]; Union[a] (* Artur Jasinski *)

Prog

(PARI) f(x)=my(L=log(x)); L/lambertw(L) \\ finds y such that y^y == x
list(lim)=my(v=List()); for(x=2, f(lim/2), my(y=x+1, t); while((t=x^y+y^x)<=lim, if(ispseudoprime(t), listput(v, t)); y+=2)); Set(v) \\ Charles R Greathouse IV, Oct 28 2014

Crossrefs

Cf. A061119 (primes where one of x,y is 2), A064539 (non-2 values where one of x,y is 2), A253471 (non-3 values where one of x,y is 3), A073499 (subset listing y where x = y+1), A076980 (Leyland numbers).

Sequence in context: A292082 A001601 A061119 * A049985 A126579 A309060

Adjacent sequences: A094130 A094131 A094132 * A094134 A094135 A094136

Keyword

nonn

Author

Lekraj Beedassy, May 04 2004

Extensions

Corrected and extended by Jens Kruse Andersen, Oct 26 2007

Edited by Hans Havermann, Apr 10 2015

Status

approved