|
%I #56 Apr 15 2022 15:47:06
%S 3,17,593,32993,2097593,8589935681,59604644783353249,
%T 523347633027360537213687137,43143988327398957279342419750374600193,
%U 4318114567396436564035293097707729426477458833,5052785737795758503064406447721934417290878968063369478337
%N Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1.
%C Contains A061119 as a subsequence.
%H 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 Ed Copeland and Brady Haran, <a href="https://www.youtube.com/watch?v=Lsu2dIr_c8k">Leyland Numbers</a>, Numberphile video (2014)
%H 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 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 Paul Leyland, <a href="http://www.leyland.vispa.com/numth/primes/xyyx.htm">Primes and PRPs of the form x^y + y^x</a>
%e 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 N:= 10^100: # to get all terms <= N
%p A:= {3}:
%p for n from 2 while 2*n^n < N do
%p for k from n+1 do if igcd(n,k)=1 then
%p a:= n^k + k^n;
%p if a > N then break fi;
%p if isprime(a) then A:= A union {a} fi fi;
%p od
%p od:
%p A; # if using Maple 11 or earlier, uncomment the next line
%p # sort(convert(A,list)); # _Robert Israel_, Apr 13 2015
%t 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_ *)
%o (PARI) f(x)=my(L=log(x)); L/lambertw(L) \\ finds y such that y^y == x
%o 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 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 nonn
%O 1,1
%A _Lekraj Beedassy_, May 04 2004
%E Corrected and extended by _Jens Kruse Andersen_, Oct 26 2007
%E Edited by _Hans Havermann_, Apr 10 2015
| 