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
Keywords
Examples
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.
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.
- Norman Luhn, Leyland table, 1st kind.
Crossrefs
Programs
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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 *) Prepend[Flatten[Map[Function[n, Map[Function[m, If[PrimeQ[m^n + n^m], m^n + n^m, Sequence[], Nothing]], Range[2, n]]], Range[2, 50]], 1], 3]//Union (* Mikk Heidemaa, Mar 27 2025 *) -
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
Extensions
Corrected and extended by Jens Kruse Andersen, Oct 26 2007
Edited by Hans Havermann, Apr 10 2015
Comments