Digital Diversity

Question

A positive integer may be represented in an integer base 1 <= b < inf.

When converted to that base it has some number of distinct digits.

Any positive integer in base 1 has 1 distinct digit.

Most positive integers in base 2 have 2 distinct digits, the exceptions being those of the form 2^n - 1, which only have 1.

So the first positive integer that may be represented in an integer base with 1 unique digit is 1 and the first that may be represented with 2 distinct digits is 2.

We can say that 1 is the first integer with digital diversity 1 and 2 is the first integer with digital diversity 2.

Challenge:

Given a positive integer n return the first positive integer (in base ten*) that has a digital diversity of n.

* if your language only supports a specific base (e.g. unary or binary) then you may output in that base.

Your algorithm must work in theory for any positive integer input: it may fail because the precision of your language's integer is too small for the output; but may not fail because base conversion is only defined up to some limit.

Test cases

input  output
   1     1
   2     2
   3     11
   4     75
   5     694
   6     8345
   7     123717
  17     49030176097150555672
  20     5271200265927977839335179
  35     31553934355853606735562426636407089783813301667210139
  63     3625251781415299613726919161860178255907794200133329465833974783321623703779312895623049180230543882191649073441
 257     87678437238928144977867204156371666030574491195943247606217411725999221158137320290311206746021269051905957869964398955543865645836750532964676103309118517901711628268617642190891105089936701834562621017362909185346834491214407969530898724148629372941508591337423558645926764610261822387781382563338079572769909101879401794746607730261119588219922573912353523976018472514396317057486257150092160745928604277707892487794747938484196105308022626085969393774316283689089561353458798878282422725100360693093282006215082783023264045094700028196975508236300153490495688610733745982183150355962887110565055971546946484175232

This is code-golf, the shortest solution in bytes wins.

OEIS: A049363 - also smallest pandigital number in base n.

Answer 1

Jelly, 4 bytes

ṖaWḅ

Try it online! or verify all test cases

How it works

ṖaWḅ  Main link. Argument: n

Ṗ     Pop; yield [1, 2, 3, ..., n-1].
  W   Wrap; yield [n].
 a    Logical AND; yield [n, 2, 3, ..., n-1].
   ḅ  Convert the result from base n to integer.

Answer 2

Python, 40 bytes

f=lambda n,k=1:n*(n<k+2)or-~f(n,k+1)*n-k

Test it on Ideone.

How it works

A number with n distinct digits must clearly be expressed in base b ≥ n. Since our goal is to minimize the number, b should also be as small as possible, so b = n is the logical choice.

That leaves us with arranging the digits 0, …, n-1 to create a number as small as possible, which means the most significant digits must be kept as small as possible. Since the first digit cannot be a 0 in the canonical representation, the smallest number is
(1)(0)(2)...(n-2)(n-1)_n = n^n-1 + 2n^n-3 + … + (n-2)n + (n-1), which f computes recursively.

Answer 3

Python 2, 54 46 bytes

This is a very very very! fast, iterative solution.

n=r=input();k=2
while k<n:r=r*n+k;k+=1
print r

Try it online

There's no recursion, so it works for large input. Here's the result of n = 17000 (takes 1-2 seconds):

http://pastebin.com/UZjgvUSW

Answer 4

JavaScript (ES6), 29 bytes

f=(b,n=b)=>n>2?f(b,--n)*b+n:b

Answer 5

J, 9 bytes

#.],2}.i.

Based on @Dennis' method.

Usage

   f =: #.],2}.i.
   (,.f"0) >: i. 7
1      1
2      2
3     11
4     75
5    694
6   8345
7 123717
   f 17x
49030176097150555672

Explanation

#.],2}.i.  Input: n
       i.  Get range, [0, 1, ..., n-1]
    2}.    Drop the first 2 values, [2, 3, ...., n-1]
  ]        Get n
   ,       Prepend it, [n, 2, 3, ..., n-1]
#.         Convert that to decimal from a list of base-n digits and return

There is an alternative solution based on using the permutation index. Given input n, create the list of digits [0, 1, ..., n], and find the permutation using an index of n!, and convert that as a list of base-n digits. The corresponding solution in J for 12 bytes

#.]{.!A.i.,]  Input: n
        i.    Make range [0, 1, ..., n-1]
           ]  Get n
          ,   Join, makes [0, 1, ..., n-1, n]
     !        Factorial of n
      A.      Permutation index using n! into [0, 1, ..., n]
  ]           Get n
   {.         Take the first n values of that permutation
              (This is to handle the case when n = 1)
#.            Convert that to decimal from a list of base-n digits and return

Answer 6

Ruby, 37 35 34 bytes

->n{s=n;(2...n).map{|d|s=s*n+d};s}

The answer for a given n takes the form 10234...(n-1) in base n. Using n=10 as an example:

Start with n: 10

Multiply by n and add 2: 102

Mutliply by n and add 3: 1023

And so on.

EDIT: Shorter to use map, it seems.

EDIT 2: Thanks for the tip, m-chrzan!

Answer 7

Haskell, 31 bytes

f n=foldl((+).(*n))n[2..n-1]

Converts the list [n,2,3,...,n-1] to base n using Horner's method via folding. A less golfed version of this is given on the OEIS page.

Thanks to nimi for 3 bytes!

Answer 8

CJam, 9 bytes

ri__,2>+b

Try it online!

Explanation

ri   e# Read input N.
__   e# Make two copies.
,    e# Turn last copy into range [0 1 2 ... N-1].
2>   e# Discard up to two leading values.
+    e# Prepend a copy of N.
b    e# Treat as base-N digits.

Answer 9

CJam (9 bytes)

qi_,X2$tb

Online demo

Dissection

Obviously the smallest number with digital diversity n is found by base-converting [1 0 2 3 ... n-1] in base n. However, note that the base conversion built-in doesn't require the digits to be in the range 0 .. n-1.

qi    e# Read integer from stdin
_,    e# Duplicate and built array [0 1 ... n-1]
X2$t  e# Set value at index 1 to n
b     e# Base conversion

Note that in the special case n = 1 we get 1 [0] 1 1 tb giving 1 [0 1] b which is 1.

Answer 10

05AB1E, 9 bytes

DL¦¨v¹*y+

Try it online!

Explanation

n = 4 used for example.

D           # duplicate input
            # STACK: 4, 4
 L          # range(1, a)
            # STACK: 4, [1,2,3,4]
  ¦¨        # remove first and last element of list
            # STACK: 4, [2,3]
    v       # for each y in list
     ¹*     # multiply current stack with input
       y+   # and add y
            # STACK, first pass: 4*4+2 = 18
            # STACK, second pass: 18*4+3 = 75

Answer 11

C++ - 181 55

Was about to post that real cool solution using <numeric>:

#import <vector>
#import <numeric>
using namespace std;int f(int n){vector<int> v(n+1);iota(v.begin(),v.end(),0);swap(v[0],v[1]);return accumulate(v.begin(),v.end()-1,0,[n](int r,int a){return r*n+a;});}

and then i realized it is way easier:

int g(int n){int r=n,j=2;for(;j<n;)r=r*n+j++;return r;}

Answer 12

Perl 6,  34 31  30 bytes

Translated from the Haskell example on the OEIS page.

{(1,0,|(2..^$^n)).reduce: $n×*+*}        # 34
{(1,0,|(2..^$^n)).reduce: $n* *+*}       # 34

{reduce $^n×*+*,1,0,|(2..^$n)}           # 31
{[[&($^n×*+*)]] 1,0,|(2..^$n)}           # 31

{reduce $_×*+*,1,0,|(2..^$_)}            # 30

• [&(…)] turns … into an in-place infix operator
• The […] shown above turns an infix op into a fold (left or right depending on the operator associativity)

Expanded:

{
  reduce

    # declare the blocks only parameter ｢$n｣ ( the ｢^｣ twigil )
    # declare a WhateverCode lambda that takes two args ｢*｣
    $^n × * + *

    # a list that always contains at least (1,0)
    1, 0,
    # with a range slipped in
    |(
      2 ..^ $n # range from 2 up-to and excluding ｢$n｣
               # it is empty if $n <= 2
    )
}

Usage:

my &code = {reduce $_×*+*,1,0,|(2..^$_)}

say code 1; # 1
say code 2; # 2
say code 3; # 11
say code 4; # 75
say code 7; # 123717

# let's see how long it takes to calculate a largish value:

my $start-time = now;
$_ = code 17000;
my $calc-time = now;
$_ = ~$_; # 25189207981120412047...86380901260421982999
my $display-time = now;

say "It takes only { $calc-time - $start-time } seconds to calculate 17000";
say "but { $display-time - $calc-time } seconds to stringify"

# It takes only 1.06527824 seconds to calculate 17000
# but 5.3929017 seconds to stringify

Answer 13

Brain-Flak, 84 76 bytes

Thanks to Wheat Wizard for golfing 8 bytes

(({})<>){(({}[()]))}{}(<{}{}>)((())){{}({<({}[()])><>({})<>}{}{})([][()])}{}

Try it online!

Explanation

The program pushes the values from 0 to n-1 to the stack replaces the top 0 and 1 with 1 and 0. Then it multiplies the top of the stack by n and adds the value below it until there is only one value remaining on the stack.

Essentially it finds the digits for the smallest number in base n that contains n different digits (for n > 1 it's always of the form 1023...(n-1)). It then calculates the number given the digits and the base.

Annotated Code

(({})<>)       # Pushes a copy of n to the right stack and switches to right stack
{(({}[()]))}{} # While the top of the stack != 0 copy the top of the stack-1
               #   and push it
(<{}{}>)       # Discard the top two values (0 and 1 for n > 1) and push 0
((()))         # Push 1 twice (second time so that the loop is works properly)
{{}            # Loop while stack height > 1
  (            #   Push...
    {<({}[()])><>({})<>}{} # The top value of the stack * n
    {}         #     Plus the value below the top of the stack
  )            #   End push
([][()])}{}    # End loop

Answer 14

PHP, 78 Bytes

for(;$i<$a=$argn;)$s=bcadd($s,bcmul($i<2?1-$i:$i,bcpow($a,$a-1-$i++)));echo$s;

Online Version

60 Bytes works only till n=16 with the precision in the testcases

For n=144 INF

n=145 NAN

for(;$j<$a=$argn;)$t+=($j<2?1-$j:$j)*$a**($a-1-$j++);echo$t;

Answer 15

Julia, 26 bytes

\(n,k=n)=k<3?n:n\~-k*n+~-k

Try it online!

Answer 16

ShapeScript, 25 bytes

_0?1'1+@2?*1?+@'2?2-*!#@#

Input is in unary, output is in decimal. Try it online!

Answer 17

k, 12 bytes

Based off of Dennis' answer.

{x/x,2+!x-2}

Answer 18

JavaScript (ES6), 39 bytes

Does not use =>

function f(b,n){throw f(b,n>2?--n:1)*b}