# Leyland Numbers

Given a natural number n, return the n-th Leyland number.

### Leyland Number

Leyland numbers are positive integers k of the form

k = x^y + y^x


Where x,y are integers strictly greater than 1.

They are enumerated in ascending order.

EDIT: @DigitalTrauma suggested I include following "definition":

Imagine we throw x^y+y^x in a bag for all possible values of x and y, and avoid throwing in duplicates. Then we sort that bag. The sorted bag is our sequence.

### Details

You may use 0 or 1 based indexing, whatever suits you best.

Your program must be able to output at least all Leyland numbers less than the maximum of signed 32-bit integers. (The last Leyland number below this limit is 1996813914, at index 82.)

### Test cases

The first few terms are following:

8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124


A076980 in OEIS, except for the first entry. Note that because of that additional first entry, the indices on OEIS are shifted by one.

More can be found in the OEIS b-file

• They are enumerated in ascending order I'm not really sure what this means. Could you provide a list of x and y? – James Jun 15 '16 at 13:26
• @DrGreenEggsandIronMan That means, 8 is before 17, not the other way round. – Leaky Nun Jun 15 '16 at 13:29
• @DrGreenEggsandIronMan Imagine we throw x^y+y^x in a bag for all possible values of x and y, and avoid thrwoing in duplicates. Then we sort that bag. The sorted bag is our sequence. – flawr Jun 15 '16 at 13:42
• Very large bag you have there – Luis Mendo Jun 15 '16 at 18:35
• @LuisMendo Ask @​HenriLéonLebesgue and he is going to tell you that this bag is basically nothing. – flawr Jun 15 '16 at 18:58

# MATL, 1615 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
u    % Keep unique elements. This linearizes (flattens) the 2D array
S    % Sort
G)   % Get the n-th entry. Implicitly display

• In Matlab unique sorts the elements. Doesn't it in MATL, too? – pajonk Jun 15 '16 at 17:20
• @pajonk MATL uses the 'stable' flag for unique by default as that is the more typical usage. – Suever Jun 15 '16 at 18:15
• @Suever Ok, thanks for clarifying. – pajonk Jun 15 '16 at 18:52
• I feel like we use the t!^ (where ^ can be replaced by +, -, or any number of operators) motif a lot. What if we made & mean 1 input for some of those where for a vector it has that behavior? – Suever Jun 15 '16 at 18:56
• @Suever That's a great idea! I've done some research with your script; see the chat – Luis Mendo Jun 15 '16 at 20:43

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..]!!)

• In hindsight such an obvious solution, I like it a lot! – flawr Jun 15 '16 at 18:52

# Java 8, 225221219216206204193 192 bytes

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


0-indexed

-2 bytes (221 → 219) saved by replacing 1996813915 with (1L<<31) thanks to @LeakyNun.
-3 bytes (219 → 216) thanks to @LeakyNun and @Frozn with something I forgot myself..
-10 bytes (216 → 206) by changing Java 7 to 8.
-2 bytes (206 → 204) by replacing ArrayList with Vector thanks to @TAsk.
-11 bytes (204 → 193) by removing s<(1L<<31)&, since the question states "at least all Leyland numbers less than the maximum of signed 32-bit integers".
-1 byte (193 → 192) by changing Vector to Stack.

Explanation:

Try it here

import java.util.*;            // Required import for Stack
n->{                           // Method with integer parameter and long return-type
List<Long>t=new Stack();     //  Temp list
for(long i=1,j,s;++i<30;)    //  Loop (1) from 2 to 30 (exclusive)
for(j=1;++j<30;){          //   Inner loop (2) from 2 to 30 (exclusive)
s=(int)(                 //    s is:
Math.pow(i,j)+Math.pow(j,i)); // i^j + j^i
if(!t.contains(s))       //     If s is not already part of the List
}                          //   End of inner loop (2)
//  End of loop (1) (implicit / single-line body)
Collections.sort(t);         //  Order the List
return t.get(n);             //  Return the item at the given index
}                              // End of method

• 10/10 for using java – Leaky Nun Jun 15 '16 at 14:17
• Since you only need to support up to 2^31-1 (i.e., signed int), can't you swap out a bunch of the long casts? – AdmBorkBork Jun 15 '16 at 14:18
• Quick golfs: import java.util.*;long c(int n){List<Long>t=new ArrayList();for(int i=2,j;i<25;i++)for(j=2;j<25;j++){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);} – Leaky Nun Jun 15 '16 at 14:26
• The for loop variable declaration. – Leaky Nun Jun 15 '16 at 14:31
• How about for (int i = 1, j; ++i < 30;) and for (j = 1; ++j < 30;) – Frozn Jun 16 '16 at 10:56

# 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.)

• Would it help to create an array of powers [[4,8,...][9,27,...]] and add it to its transpose? – Neil Jun 15 '16 at 14:05
• @Neil I don't think so. It would be helpful in Jelly, but not in Pyth. Pyth does not automatically vectorize. – Leaky Nun Jun 15 '16 at 14:12
• Also helps in MATL, it seems. – Neil Jun 15 '16 at 19:44
• Why do you keep the slower version? – Erik the Outgolfer Jun 16 '16 at 14:18

# 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

ÝÌ2ãvyÂmsm+}){Ù¹è


Explained

ÝÌ                     # range(2,N+2)
2ã                   # get all pairs of numbers in the range
v                  # for each pair
yÂmsm+          # 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

• Very nice! One tip, ÝÌ is short for >L>. – Adnan Jun 15 '16 at 15:50
• @Adnan: Thanks! I can't belive I didn't think of that :P – Emigna Jun 15 '16 at 15:54
• ê is sorted_uniquified, if that existed when this was asked. – Magic Octopus Urn Nov 1 '16 at 16:36
• @carusocomputing: It was bugged until quite recently I'm afraid. – Emigna Nov 1 '16 at 18:07

# Mathematica, 6048 40 bytes

(Union@@Array[#^#2+#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.

Saved 8 bytes thanks to @LLlAMnYP.

## Usage

• {#+1,#+1} isn't necessary, can be left as {#,#} and {2,2} can be replaced with simply 2. – LLlAMnYP Jun 17 '16 at 11:20
• @LLlAMnYP Thanks! Didn't know that Array would expand the third argument. – miles Jun 17 '16 at 11:42
• Neither did I but I decided to try it anyway and it worked :) – LLlAMnYP Jun 17 '16 at 11:44

# 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)

2r30*€¹$+Z$FṢQị@


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

• R‘*€¹$+Z$FṢQị@ is faster, shorter and has no artificial upper bound. – Dennis Jun 15 '16 at 15:31
• @Dennis and beats my answer :-P – Luis Mendo Jun 15 '16 at 15:31
• @Dennis I don't get it. How come it is faster than the second one. – Leaky Nun Jun 15 '16 at 15:32
• It isn't faster than the second one. The execution time is too short to get an accurate measurement. – Dennis Jun 15 '16 at 15:34
• Now 13 bytes :-P – Luis Mendo Jun 15 '16 at 15:51

printf %s\\n x={2..32}\;y={2..32}\;x^y+y^x|bc|sort -nu|sed $1!d  1-based indexing. It looks like this is pretty much the same approach as @TimmyD's answer. Instead of nested loops, bash brace expansion is used to generate arithmetic expressions that are piped to bc for evaluation. Ideone. # Perl 6, 60 58 56 bytes {sort(keys bag [X[&({$^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
)[$_] }  • Can't you remove the spaces between sort [ and ] 2..31? – Erik the Outgolfer Jun 16 '16 at 14:10 • @EʀɪᴋᴛʜᴇGᴏʟғᴇʀ That would turn it from a subroutine call sort([... to an array access of a term sort[.... A similar thing happens with the other space. – Brad Gilbert b2gills Jun 16 '16 at 16:41 # F#, 117, 104 Welp, it's shorter than my C# answer at least. Saved 13 bytes thanks to Reed Copsey in the F# chatroom. let f n=[for x in 2I..32I do for y in 2I..32I->x**(int y)+y**(int x)]|>Seq.sort|>Seq.distinct|>Seq.nth n  ## PowerShell v2+, 8473 68 bytes (2..30|%{2..($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.

g n=(f.toInteger$n+3)!!n  • import Data.List f n|w<-[2..toEnum$n+3]=(sort$nub[x^y+y^x|x<-w,y<-w])!!n Do you know why toInteger/toEnum is needed? – Damien Jun 15 '16 at 16:20 • Wow, this is crazy=) Feel free to add it as your own answer, as it is qutie different from mine! If we omit toInteger in my solution we'll have an overflow using int, because we iterate way higher (to n+3 instead of n) when working with the list. Otherwise we'd need to hardcode the first four terms or so. What exactly does toEnum do in your solution? – flawr Jun 15 '16 at 16:38 • OK, that's because of (!!) operator which binds n to an Int. Since n is supposed to be under 82, w can be replaced by [2..99] for example and f=(sort(nub[x^y+y^x|x<-[2..99],y<-[2..x]])!!) . toEnum converts an Int to an Enum, and Integer is an instance of Enum class so toEnum here converts n+3 to an Integer. – Damien Jun 15 '16 at 16:42 ## 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 • It’s okay to just write your answer as r=range(2,32) lambda n:sorted(…)[n] – Lynn Jun 15 '16 at 16:19 # 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];  • Still shorter than Java. – flawr Jun 15 '16 at 19:06 • @flawr Wooooooo? – Morgan Thrapp Jun 15 '16 at 19:07 • I know nothing about C#, but couldn't you save Enumerable.Range( to a variable/function/iterator/whatever with a shorter name for reuisng? – flawr Jun 15 '16 at 19:10 • I could, but then I would need to include a class and type defs, which ends up costing me a ton. – Morgan Thrapp Jun 15 '16 at 19:10 # 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

# J, 29 bytes

<:{[:/:~@~.@,[:(^+^~)"0/~2+i.


Uses one-based indexing. Conversion from my Mathematica solution.

The true secret here is that I have :(^+^~) on my side.

## Usage

   f =: <:{[:/:~@~.@,[:(^+^~)"0/~2+i.
f 7
145
(,.f"0) >: i. 10  NB. Extra commands for formatting
1   8
2  17
3  32
4  54
5  57
6 100
7 145
8 177
9 320
10 368


## Explanation

<:{[:/:~@~.@,[:(^+^~)"0/~2+i.  Input: n
2+i.  Step one
"0/~      Step two
:(^+^~)          ???
<:{[:/:~@~.@,[                 Profit


More seriously,

<:{[:/:~@~.@,[:(^+^~)"0/~2+i.  Input: n
i.  Create the range [0, 1, ..., n-1]
(^+^~)"0        Create a dyad (2 argument function) with inputs x, y
and returns x^y + y^x
[:        /~      Use that function to create a table using the previous range
[:       ,                  Flatten the table into a list
~.@                   Take its distinct values only
/:~@                      Sort it in ascending order
<:                             Decrement n (since J is zero-indexed)
{                            Select the value at index n-1 from the list and return

• ... Profit :D – flawr Jun 17 '16 at 19:19

# Swift 3, 138 bytes

import Glibc;func l(n:Int)->Int{let r=stride(from:2.0,to:50,by:1);return Int(Set(r.flatMap{x in r.map{pow(x,$0)+pow($0,x)}}).sorted()[n])}


Ungolfed code

Try it here

import Glibc
func l(n: Int) -> Int {
// Create a Double sequence from 2 to 50 (because pow requires Double)
let r = stride(from: 2.0, to: 50.0, by: 1.0)

return Int(Set(r.flatMap {
x in r.map {
pow(x, $0) + pow($0, x)
}
}).sorted()[n])

• Welcome to Programming Puzzles and Code Golf! Nice first answer, but it'd be better if you could explain what's going on. – clismique Jun 16 '16 at 9:08

Axiom 148 bytes

w(n)==(v:List INT:=[];for i in 2..31 repeat for j in i..31 repeat(a:=i^j+j^i;if a>1996813914 then break;v:=cons(a,v));v:=sort v;~index?(n,v)=>0;v.n)


some example

w(n)==
v:List INT:=[];for i in 2..31 repeat for j in i..31 repeat
(a:=i^j+j^i;if a>1996813914 then break;v:=cons(a,v));v:=sort v;~index?(n,v)=>0
v.n
(2) -> [w(i)  for i in 0..85]
Compiling function w with type NonNegativeInteger -> Integer

(2)
[0, 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124, 1649,
2169, 2530, 4240, 5392, 6250, 7073, 8361, 16580, 18785, 20412, 23401,
32993, 60049, 65792, 69632, 93312, 94932, 131361, 178478, 262468, 268705,
397585, 423393, 524649, 533169, 1048976, 1058576, 1596520, 1647086,
1941760, 2012174, 2097593, 4194788, 4208945, 4785713, 7861953, 8389137,
9865625, 10609137, 14352282, 16777792, 16797952, 33554432, 33555057,
43050817, 45136576, 48989176, 61466176, 67109540, 67137425, 129145076,
134218457, 177264449, 244389457, 268436240, 268473872, 292475249,
364568617, 387426321, 536871753, 774840978, 1073742724, 1073792449,
1162268326, 1173741824, 1221074418, 1996813914, 0, 0, 0]


# Perl 5, 70 + 1 (-p) = 71 bytes

for$x(2..32){$r{$x**$_+$_**$x}++for$x..32}$_=(sort{$a<=>$b}keys%r)[\$_]


Try it online!

# Ruby, 62 58 bytes

->n{a,*b=c=2;c+=1while(b<<a**c+c**a;c<32||32>c=a+=1);b[n]}


Try it online!

# J, 38 31 bytes

0-indexed.

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


## Usage

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


# 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.

# Python 3, 129->116 bytes

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

t=[]
for k in range(100):a,b,q=k//10,k%10,a**b+b**a;f=lambda q:0if 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

• Make t a set instead of a list, and replace the last for statements with a plain t.add(q). – Cristian Ciupitu Jun 17 '16 at 0:37

# APL (Dyalog), 27 bytes

{d⊃⍨⍵⊃⍋d←∪,∘.(*⍨+*)⍨1↓⍳1+⍵}


Try it online!

# Japt -g, 15 bytes

g2ôU ïÈ**Y+pXÃü


Try it

g2ôU ïÈ**Y+pXÃü
:Implicit input of integer U
g                   :Index into the following array
2ôU                :  Range [2,U+2]
ï              :  Cartesian product with itself
È             :  Reduce each pair [X,Y]
**Y          :    Raise X to the power of Y
+pX       :    Add Y raised to the power of X
Ã      :  End reduce
ü     :  Sort & partition by value
:Implicit output of first element