MATL, 16 15 13 bytes

Q:Qt!^t!+uSG)

Output is 1-based.

Try it online!

Explanation

Q    % Take input n. Add 1
:Q   % Range [2 ... n+1]. This is enough to be used as x and y
t!   % Duplicate and transpose
^    % Power, element-wise with broadcast. Gives 2D, square array with x^y
     % for all pairs of x and y
t!   % Duplicate and transpose. Gives square array with y^x
+    % Add, element-wise
u    % Keep unique elements. This linearizes (flattens) the 2D array
S    % Sort
G)   % Get the n-th entry. Implicitly display

Java 7, 225 221 219 bytes

import java.util.*;long c(int n){List<Long>t=new ArrayList();for(int i=2,j;++i<30;)for(j=2;++j<30;){long s=(long)(Math.pow(i,j)+Math.pow(j,i));if(s<(1L<<31)&!t.contains(s))t.add(s);}Collections.sort(t);return t.get(n);}

2 bytes saved by replacing 1996813915 with (1L<<31) thanks to @LeakyNun
(and 4 more thanks to @LeakyNun and @Frozn with something I forgot myself..)

Ungolfed & test code:

Try it here

import java.util.*;

public class Main{
  static long c(int n){
    List<Long> t = new ArrayList();
    for(int i = 2, j; ++i < 30; ){
      for(j = 2; ++j < 30; ){
        long s = (long) (Math.pow(i, j) + Math.pow(j, i));
        if (s < (1L<<31) & !t.contains(s)){
          t.add(s);
        }
      }
    }
    Collections.sort(t);
    return t.get(n);
  }

  public static void main(String[] a) {
    // Test code for altering output (displaying one, skipping one, etc.)
    for (int i = 0; i < 14; i += 2) {
      System.out.println(c(i));
    }
  }
}

Output:

8
32
57
145
320
512
945

Pyth, 17 bytes

0-indexed.

@{Sms^M_Bd^}2+2Q2

Try it online! (Please, keep it at 100.)

How it works

@{Sms^M_Bd^}2+2Q2
@{Sms^M_Bd^}2+2Q2Q  implicit filling. Input:Q

           }2+2Q    Yield the array [2,3,4,...,Q+2]
          ^     2   Cartesian square: yield all the
                    pairs formed by the above array.
   m     d          Map the following to all of those pairs (e.g. [2,3]):
       _B               Create [[2,3],[3,2]]
     ^M                 Reduce by exponent to each array:
                        create [8,9]
    s                   Sum:   17     (Leyland number generated)
  S                 Sort the generated numbers
 {                  Remove duplicate
@                Q  Find the Q-th element.

Slower version

1-indexed.

e.ffqZs^M_BT^}2Z2

Try it online! (Please, keep it at 3.)

Haskell, 52 bytes

r=[2..31]
([k|k<-[0..],elem k[x^y+y^x|x<-r,y<-r]]!!)

Really inefficient. Tests each natural number for being a Leyland number, making an infinite list of those that are. Given an input, takes that index element of the list. Uses that only x,y up to 31 need to be checked for 32 bit integers.

Same length with filter:

r=[2..31]
(filter(`elem`[x^y+y^x|x<-r,y<-r])[0..]!!)

MATLAB, 58 bytes

1-indexed

n=input('');[A B]=ndgrid(2:n+9);k=A.^B;x=unique(k'+k);x(n)

unique in MATLAB flattens and sorts the matrix.

Thanks for help to @FryAmTheEggman and @flawr.

05AB1E, 20 19 bytes

0-indexed

ÝÌ2ãvyÂ`ms`m+}){Ù¹è

Explained

ÝÌ                     # range(2,N+2)
  2ã                   # get all pairs of numbers in the range
    v                  # for each pair
     yÂ`ms`m+          # push x^y+y^x
             }         # end loop
              ){Ù      # wrap to list, sort and remove duplicates
                 ¹è    # get Nth element of list

Try it online

Saved 1 byte thanks to @Adnan

Bash + GNU utilities, 63

printf %s\\n x={2..32}\;y={2..32}\;x^y+y^x|bc|sort -nu|sed $1!d

1-based indexing. It looks like this is pretty much the same approach as @TimmyD's answer. Instead of nested loops, bash brace expansion is used to generate arithmetic expressions that are piped to bc for evaluation.

Ideone.

PowerShell v2+, 84 73 68 bytes

(2..30|%{2..($x=$_)|%{"$x*"*$_+'1+'+"$_*"*$x+1|iex}}|sort)[$args[0]]

Saved 11 bytes thanks to @Neil ... saved additional 5 bytes by reorganizing how the iex expression is evaluated.

Naïve method, we simply double-for loop from x=2..30 and y=2..x. Each loop we put x^y + y^x on the pipeline. The 30 was chosen experimentally to ensure that we covered all cases less than 2^31-1 ;-). We pipe those to Sort-Object to order them ascending. Output is zero-indexed based on the input $args[0].

Yes, there are a lot of extraneous entries generated here -- this algorithm actually generates 435 Leyland numbers -- but things above index 81 are not guaranteed to be accurate and in order (there may be some that are skipped).

Examples

PS C:\Tools\Scripts\golfing> .\leyland-numbers.ps1 54
14352282

PS C:\Tools\Scripts\golfing> .\leyland-numbers.ps1 33
178478

PS C:\Tools\Scripts\golfing> .\leyland-numbers.ps1 77
1073792449

Jelly, 14 bytes

2 bytes thanks to Dennis.

R‘*€¹$+Z$FṢQị@

Try it online! (Takes ~ 1s for 82 for me) (O(n^2) time)

Original 16-byte answer

2r30*€¹$+Z$FṢQị@

Try it online! (Takes < 1s for me) (Constant time)

Haskell, 99 98 96 95 94 bytes

It is probably easily outgolfed, but that was the best I was able to come up with.

import Data.List
f n|n<2=[]|n>1=sort.nub$f(n-1)++[x^n+n^x|x<-[2..n]]
g n=(f.toInteger$n+3)!!n

Python 3, 76 69 bytes

r=range(2,32);f=lambda n:sorted({x**y+y**x for x in r for y in r})[n]

0-indexed.

https://repl.it/C2SA

Perl 6,  60 58  56 bytes

{sort(keys bag [X[&({$^a**$^b+$b**$a})]] (2..$_+2)xx 2)[$_]}
{sort(keys set [X[&({$^a**$^b+$b**$a})]] (2..$_+2)xx 2)[$_]}
{sort(unique [X[&({$^a**$^b+$b**$a})]] (2..$_+2)xx 2)[$_]}
{squish(sort [X[&({$^a**$^b+$b**$a})]] (2..$_+2)xx 2)[$_]}
{squish(sort [X[&({$^a**$^b+$b**$a})]] (2..31)xx 2)[$_]}
{squish(sort [X[&({$^a**$^b+$b**$a})]] 2..31,2..31)[$_]}

Test:

#! /usr/bin/env perl6
use v6.c;

my &Leyland = {squish(sort [X[&({$^a**$^b+$b**$a})]] 2..31,2..31)[$_]}

say ^14 .map: &Leyland;
time-this {Leyland 81};

sub time-this (&code) {
  my $start = now;
  my $value = code();
  printf "takes %.3f seconds to come up with $value\n", now - $start;
}

(8 17 32 54 57 100 145 177 320 368 512 593 945 1124)
takes 0.107 seconds to come up with 1996813914

Explanation:

{
  squish( # remove repeated values
    sort
      [X[&( # cross reduce with:
        { $^a ** $^b + $b ** $a }
      )]]
        ( 2 .. $_+2 ) # ｢Range.new(2,$_+2)｣ (inclusive range)
        xx 2          # repeat list
  )[$_]
}

JavaScript (Firefox 42+), 94 bytes

n=>[for(x of Array(32).keys())for(y of Array(x+1).keys())if(y>1)x**y+y**x].sort((x,y)=>x-y)[n]

Needs Firefox 42 because it uses both array comprehensions and exponentiation ([for(..of..)] and **).

C#, 141, 127 bytes.

Oh c#, you are such a long language.

n=>(from x in Enumerable.Range(2,32)from y in Enumerable.Range(2,32)select Math.Pow(x,y)+Math.Pow(y,x)).Distinct().ToList()[n];

This is a lambda that needs to be assigned to delegate double del(int n); to be run, as such:

delegate double del(int n);
del f=n=>(from x in Enumerable.Range(2,32)from y in Enumerable.Range(2,32)select Math.Pow(x,y)+Math.Pow(y,x)).OrderBy(q=>q).Distinct().ToList()[n];

F#, 117, 104

Welp, it's shorter than my C# answer at least.

Saved 13 bytes thanks to Reed Copsey in the F# chatroom.

let f n=[for x in 2I..32I do for y in 2I..32I->x**(int y)+y**(int x)]|>Seq.sort|>Seq.distinct|>Seq.nth n

SQL (PostgreSQL 9.4), 171 bytes

Done as a prepared statement. Generate a couple of series 2 - 99, cross join them and do the equation. Densely rank the results to index them and select the first result that has the rank of the integer input.

prepare l(int)as select s from(select dense_rank()over(order by s)r,s from(select x^y+y^x from generate_series(2,99)x(x),generate_series(2,99)y(y))c(s))d where r=$1limit 1

Executed as follows

execute l(82)
s
-----------------
1996813914

This ended up running a lot quicker than I expected

Mathematica, 60 48 bytes

(Union@@Array[#^#2+#2^#&,{#+1,#+1},{2,2}])[[#]]&

Uses one-based indexing. Union is used by applying it between each row of the 2D matrix created by the Array. There, Union will flatten the 2D matrix into a list while also removing any duplicates and placing the values in sorted order.

Usage

J, 38 31 bytes

0-indexed.

[{[:(#~~:)@/:~@,/[:(+|:)[:^/~2+i.@>:@]
((#~~:)/:~,/(+|:)^/~2+i.29x){~[

Usage

>> f =: ((#~~:)/:~,/(+|:)^/~2+i.29x){~[
>> f 81
<< 1996813914

Python 3, 129 bytes

I know there is a shorter python 3 answer, but I still wanted to contribute my solution.

m=n=10;t=[]
for k in range(n*m):a,b=(-~k)//n,(-~k)%m;q=a**b+b**a;f=lambda q: 0 if q in t else t.append(q);f(q)
print(sorted(t)[7:])

This was the best way that I could to think of to handle going through all values for x and all values for y. If anyone can golf my approach it would be appreciated

Swift 3, 146 bytes

import Glibc;func l(n:Int)->Int{let r=stride(from:2.0,to:15,by:1);let v=Set(r.flatMap{x in r.map{Int(pow(x,$0)+pow($0,x))}}).sorted();return v[n]}

Ungolfed code

Try it here

import Glibc
func l(n: Int) -> Int {
    let r = stride(from:2.0, to: 15.0, by: 1.0)
    let v = Set(r.flatMap {x in r.map { Int(pow(x, $0) + pow($0, x)) } }).sorted()
    return v[n]
}

Java, 200 197 bytes

0-indexed

n->{long[]t=new long[999];for(int i=1,j;++i<31;)for(j=1;j++<i;)t[i*31+j]=(long)(Math.pow(i,j)+Math.pow(j,i));return java.util.Arrays.stream(t).sorted().distinct().skip(n+1).findAny().getAsLong();};

Looks like java's streams can actually save bytes! Who would've thought?!

Ungolfed:

package pcg;

import java.util.function.IntToLongFunction;

public class Pcg82981 {

  public static void main(String[] args) {
    IntToLongFunction f = (n) -> {
      long[] t = new long[999];
      for (int i = 1; ++i < 31; ) {
        for (int j = 1; j++ < i; ) {
          t[i * 31 + j] = (long) (Math.pow(i, j) + Math.pow(j, i));
        }
      }
      return java.util.Arrays.stream(t)
          .sorted()
          .distinct()
          .skip(n + 1) // We don't care about 0.
          .findAny()   // equivalent to `findFirst`
          .getAsLong();
    };

    for (int i = 0; i < 82; i++) {
      System.out.println(f.applyAsLong(i));
    }

  }
}

Edits:

1. 200 -> 197 : removed space after long[] and removed parenthesis around n.