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

%I A094133 #67 Mar 27 2025 10:10:17
%S A094133 3,17,593,32993,2097593,8589935681,59604644783353249,
%T A094133 523347633027360537213687137,43143988327398957279342419750374600193,
%U A094133 4318114567396436564035293097707729426477458833,5052785737795758503064406447721934417290878968063369478337
%N A094133 Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1.
%C A094133 Contains A061119 as a subsequence.
%H A094133 Charles R Greathouse IV and Hans Havermann (Charles R Greathouse IV to 49), <a href="/A094133/b094133.txt">Table of n, a(n) for n = 1..100</a>
%H A094133 Ed Copeland and Brady Haran, <a href="https://www.youtube.com/watch?v=Lsu2dIr_c8k">Leyland Numbers</a>, Numberphile video (2014).
%H A094133 Hans Havermann, <a href="http://chesswanks.com/num/a094133.txt">Table of n (where known), Leyland index, number of digits in decimal representation, and (x,y) pair for all known solutions</a>.
%H A094133 Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H A094133 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/primes/xyyx.htm">Primes and PRPs of the form x^y + y^x</a>.
%H A094133 Norman Luhn, <a href="https://pzktupel.de/Primetables/TableLeyland1.php">Leyland table, 1st kind</a>.
%e A094133 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.
%p A094133 N:= 10^100: # to get all terms <= N
%p A094133 A:= {3}:
%p A094133 for n from 2 while 2*n^n < N do
%p A094133   for k from n+1 do if igcd(n,k)=1 then
%p A094133      a:= n^k + k^n;
%p A094133      if a > N then break fi;
%p A094133      if isprime(a) then A:= A union {a} fi fi;
%p A094133   od
%p A094133 od:
%p A094133 A; # if using Maple 11 or earlier, uncomment the next line
%p A094133 # sort(convert(A,list)); # _Robert Israel_, Apr 13 2015
%t A094133 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_ *)
%t A094133 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 *)
%o A094133 (PARI) f(x)=my(L=log(x)); L/lambertw(L) \\ finds y such that y^y == x
%o A094133 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
%Y A094133 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).
%K A094133 nonn
%O A094133 1,1
%A A094133 _Lekraj Beedassy_, May 04 2004
%E A094133 Corrected and extended by _Jens Kruse Andersen_, Oct 26 2007
%E A094133 Edited by _Hans Havermann_, Apr 10 2015