\$n\$-perfect numbers

Question

A positive integer \$x\$ is an \$n\$-perfect number if \$\sigma(x) = nx\$, where \$\sigma(x)\$ is the divisor sum function. For example, \$120\$ is a \$3\$-perfect number because its divisors sum to \$360\$:

$$360 = 3\times120 = 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120$$

and

$$926073336514623897600 = 6\times154345556085770649600 = 1+2+3+4+5+6+7+8+9+10+11+12+\dots+51448518695256883200+77172778042885324800+154345556085770649600$$

so \$154345556085770649600\$ is a \$6\$-perfect number.

You are to take an integer \$x\$ as input and output a value \$n\$, such that \$x\$ is an \$n\$-perfect number. If no such \$n\$ exists, you may output any consistent value that isn't a positive integer. You will never receive an input outside the bounds of your language, but your algorithm must work for arbitrarily large \$x\$.

This is code-golf so the shortest code in bytes wins.

_{Mini challenge: Beat 5 bytes in Jelly}

Test cases

    x -> n
    1 -> 1
    2 -> 0
    3 -> 0
    4 -> 0
    5 -> 0
    6 -> 2
   28 -> 2
  120 -> 3
  496 -> 2
  500 -> 0
  672 -> 3
30240 -> 4
154345556085770649600 -> 6

Answer 1

Jelly, 5 bytes

R×iÆs

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Explanation:

      - Explanation (sample for input 6)
R     - Range ([1, 2, 3, 4, 5, 6])
 ×    - Multiply by input ([6, 12, 18, 24, 30, 36])
   Æs - Divisor sum (12)
  i   - Index of divisor sum in list, else 0 (2)

Answer 2

Husk, 4 bytes

¦¹ΣḊ

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The last test case times out.

Explanation

¦¹ΣḊ   Input is a number x.
   Ḋ   List of divisors.
  Σ    Sum.
¦      Division if divisible, 0 if not
 ¹     by x.

¦ is usually just a divisibility test, but here its return value is useful.

Answer 3

Brachylog, 6 bytes

f+;?/ℕ

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How it works

f+;?/ℕ
f+     the sum of the factors
  ;?/ℕ divided by the input
     ℕ is a natural number

Alternative version, I think is cooler, but longer:

f+~×[?,.]∧
f+          the sum of the factors
  ~×        unifies with the multiplication of
    [?,.]   the input and the output
         ∧  return the output

Answer 4

05AB1E, 7 6 bytes

Ý*IÑOk

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Explanation:

Ý*IÑOk>
Ý        0-Index inclusive range of input (6 -> [1, 2, 3, 4, 5, 6])
 *       Multiply by input ([6, 12, 18, 24, 30, 36])
  IÑO    Get input -> divisors -> sum (6 -> [1, 2, 3, 6] -> 12)
     k   0-Index of divisor-sum in array or -1 if not found. ([6, >12<, 18, 24, 30, 36] -> 1)

I just used Sisyphus' method. This could probably be golfed down or even made more efficient, but I lack the 05AB1E knowledge to do so. Just thought I'd give it a shot to pass the time.

-1 Byte thanks to ovs

Answer 5

C (gcc), 47 bytes

s,i;f(x){for(i=s=x;--i;)s+=x%i?0:i;s/=s%x*s+x;}

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Returns n or 0.

Answer 6

APL (Dyalog Unicode), 16 15 13 bytes

Thanks to Bubbler for pointing out I can change the output format to save a couple of bytes

⍸×∘⍳⍨=1⊥∘∪⊢∨⍳

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Outputs a singleton list of n when n exists, and an empty array otherwise. Finds the index of (⍸) where the sum of (1⊥) the divisors (∪⊢∨⍳) equals (=) a multiple of the input (×∘⍳⍨). I use ⍸ and = instead of just ⍳ to find the index because it returns an empty list when the element isn't there rather than the length of the list.

Answer 7

Whitespace, 153 bytes

[S S S N
_Push_0][S N
S _Duplicate_0][T   N
T   T   _Read_STDIN_as_integer][T   T   T   _Retrieve_input][S N
S _Duplicate_input][S N
S _Duplicate_input][N
S S N
_Create_Label_LOOP][S S S T N
_Push_1][T  S S T   _Subtract][S N
S _Duplicate][N
T   S S N
_If_0_Jump_to_Label_REACHED_ZERO][S T   S S T   S N
_Copy_0-based_2nd_input][S T    S S T   N
_Copy_0-based_1st_integer][T    S T T   _Modulo][N
T   S T N
_If_0_Jump_to_Label_ADD_TO_SUM][N
S N
N
_Jump_to_Label_LOOP][N
S S T   N
_Create_Label_ADD_TO_SUM][S N
T   _Swap_top_two][S T  S S T   N
_Copy_0-based_1st_integer][T    S S S _Add_top_two][S N
T   _Swap_top_two][N
S N
N
_Jump_to_Label_LOOP][N
S S S N
_Create_Label_REACHED_ZERO][S N
N
_Discard_top][S N
S _Duplicate_top][S T   S S T   S N
_Copy_0-based_2nd_input][T  S T T   _Modulo][N
T   S S S N
_If_0_Jump_to_Label_DIVISIBLE][S S S N
_Push_0][N
S N
S T 
_Jump_to_Label_OUTPUT][N
S S S S N
_Create_Label_DIVISIBLE][S N
T   _Swap_top_two][T    S T S _Integer_divide_top_two][N
S S S T N
_Create_Label_OUTPUT][T N
S T _Output_as_integer]

Letters S (space), T (tab), and N (new-line) added as highlighting only.
[..._some_action] added as explanation only.

Try it online (with raw spaces, tabs and new-lines only).

Explanation in pseudo-code:

Integer input = STDIN as input
Integer sum = input
Integer i = input
Start LOOP:
  i = i - 1
  If(i == 0):
    Jump to Label REACHED_ZERO
  If(input % i == 0):
    sum = sum + i
  Go to next iteration of LOOP

Label REACHED_ZERO:
  Integer output
  If(sum % input == 0):
    output = sum integer-divided by input
  Else:
    output = 0

  Print output as integer to STDOUT

Example run: input = 6

Command    Explanation                       Stack         Heap   STDIN  STDOUT  STDERR

SSSN       Push 0                            [0]
SNS        Duplicate top (0)                 [0,0]
TNTT       Read STDIN as integer             [0]           {0:6}  6
TTT        Retrieve at address (0)           [6]           {0:6}
SNS        Duplicate top (6)                 [6,6]         {0:6}
SNS        Duplicate top (6)                 [6,6,6]       {0:6}
NSSN       Create Label LOOP                 [6,6,6]       {0:6}
 SSSTN     Push 1                            [6,6,6,1]     {0:6}
 TSST      Subtract top two (6-1)            [6,6,5]       {0:6}
 SNS       Duplicate top (5)                 [6,6,5,5]     {0:6}
 NTSSN     If 0: Jump to Label REACHED_ZERO  [6,6,5]       {0:6}
 STSSTSN   Copy 0-based 2nd (6)              [6,6,5,6]     {0:6}
 STSSTN    Copy 0-based 1st (5)              [6,6,5,6,5]   {0:6}
 TSTT      Modulo top two (6%5)              [6,6,5,1]     {0:6}
 NTSTN     If 0: Jump to Label ADD_TO_SUM    [6,6,5]       {0:6}
 NSNN      Jump to Label LOOP                [6,6,5]       {0:6}

 SSSTN     Push 1                            [6,6,5,1]     {0:6}
 TSST      Subtract top two (5-1)            [6,6,4]       {0:6}
 SNS       Duplicate top (4)                 [6,6,4,4]     {0:6}
 NTSSN     If 0: Jump to Label REACHED_ZERO  [6,6,4]       {0:6}
 STSSTSN   Copy 0-based 2nd (6)              [6,6,4,6]     {0:6}
 STSSTN    Copy 0-based 1st (4)              [6,6,4,6,4]   {0:6}
 TSTT      Modulo top two (6%4)              [6,6,4,2]     {0:6}
 NTSTN     If 0: Jump to Label ADD_TO_SUM    [6,6,4]       {0:6}
 NSNN      Jump to Label LOOP                [6,6,4]       {0:6}

 SSSTN     Push 1                            [6,6,4,1]     {0:6}
 TSST      Subtract top two (4-1)            [6,6,3]       {0:6}
 SNS       Duplicate top (3)                 [6,6,3,3]     {0:6}
 NTSSN     If 0: Jump to Label REACHED_ZERO  [6,6,3]       {0:6}
 STSSTSN   Copy 0-based 2nd (6)              [6,6,3,6]     {0:6}
 STSSTN    Copy 0-based 1st (3)              [6,6,3,6,3]   {0:6}
 TSTT      Modulo top two (6%3)              [6,6,3,0]     {0:6}
 NTSTN     If 0: Jump to Label ADD_TO_SUM    [6,6,3]       {0:6}
 NSSTN     Create Label ADD_TO_SUM           [6,6,3]       {0:6}
  SNT      Swap top two                      [6,3,6]       {0:6}
  STSSTN   Copy 0-based 1st (3)              [6,3,6,3]     {0:6}
  TSSS     Add top two (6+3)                 [6,3,9]       {0:6}
  SNT      Swap top two                      [6,9,3]       {0:6}
  NSNN     Jump to Label LOOP                [6,9,3]       {0:6}

 SSSTN     Push 1                            [6,9,3,1]     {0:6}
 TSST      Subtract top two (3-1)            [6,9,2]       {0:6}
 SNS       Duplicate top (2)                 [6,9,2,2]     {0:6}
 NTSSN     If 0: Jump to Label REACHED_ZERO  [6,9,2]       {0:6}
 STSSTSN   Copy 0-based 2nd (6)              [6,9,2,6]     {0:6}
 STSSTN    Copy 0-based 1st (5)              [6,9,2,6,2]   {0:6}
 TSTT      Modulo top two (6%5)              [6,9,2,0]     {0:6}
 NTSTN     If 0: Jump to Label ADD_TO_SUM    [6,9,2]       {0:6}
  SNT      Swap top two                      [6,2,9]       {0:6}
  STSSTN   Copy 0-based 1st (2)              [6,2,9,2]     {0:6}
  TSSS     Add top two (9+2)                 [6,2,11]      {0:6}
  SNT      Swap top two                      [6,11,2]      {0:6}
  NSNN     Jump to Label LOOP                [6,11,2]      {0:6}

 SSSTN     Push 1                            [6,11,2,1]    {0:6}
 TSST      Subtract top two (2-1)            [6,11,1]      {0:6}
 SNS       Duplicate top (1)                 [6,11,1,1]    {0:6}
 NTSSN     If 0: Jump to Label REACHED_ZERO  [6,11,1]      {0:6}
 STSSTSN   Copy 0-based 2nd (6)              [6,11,1,6]    {0:6}
 STSSTN    Copy 0-based 1st (1)              [6,11,1,6,1]  {0:6}
 TSTT      Modulo top two (6%1)              [6,11,1,0]    {0:6}
 NTSTN     If 0: Jump to Label ADD_TO_SUM    [6,11,1]      {0:6}
  SNT      Swap top two                      [6,1,11]      {0:6}
  STSSTN   Copy 0-based 1st (1)              [6,1,11,1]    {0:6}
  TSSS     Add top two (11+1)                [6,1,12]      {0:6}
  SNT      Swap top two                      [6,12,1]      {0:6}
  NSNN     Jump to Label LOOP                [6,12,1]      {0:6}

 SSSTN     Push 1                            [6,12,1,1]    {0:6}
 TSST      Subtract top two (1-1)            [6,12,0]      {0:6}
 SNS       Duplicate top (1)                 [6,12,0,0]    {0:6}
 NTSSN     If 0: Jump to Label REACHED_ZERO  [6,12,0]      {0:6}
 NSSSN     Create Label REACHED_ZERO         [6,12,0]      {0:6}
  SNN      Discard top (0)                   [6,12]        {0:6}
  SNS      Duplicate top (12)                [6,12,12]     {0:6}
  STSSTSN  Copy 0-based 2nd (6)              [6,12,12,6]   {0:6}
  TSTT     Modulo top two (12%6)             [6,12,0]      {0:6}
  NTSSSN   If 0: Jump to Label DIVISIBLE     [6,12]        {0:6}
  NSSSSN   Create Label DIVISIBLE            [6,12]        {0:6}
   SNT     Swap top two                      [12,6]        {0:6}
   TSTS    Integer-divide top two (12/6)     [2]           {0:6}
   NSSSTN  Create Label OUTPUT               [2]           {0:6}
    TNST   Output top as integer (2)         []            {0:6}         2
                                                                                 error

Stops with an error after printing the result, because no exit is defined.

Answer 8

R, 42 41 39 bytes

Edit: -1 byte (and, inspired by this, -2 more bytes) thanks to Robin Ryder

function(x)(d=sum(1:x*!x%%1:x))/x*!d%%x

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Commented:

perfect_n=
function(x)
 (d=            # d is the divisor sum, calculated as...
  sum(          # sum of...
   1:x*         # the values of 1..x that have...
    !           # zero values for...
     x%%1:x)    # x MOD 1..x
  )
 )/x            # output d/x...
    *!d%%x      # but only if it's an integer 
                # (so d MOD x == 0)

Answer 9

Pyth, 15 bytes

&!%Jsf!%QTSQQ/J

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Explanation

&!%Jsf!%QTSQQ/J
   J              # set J to
    s             # sum of
     f    SQ      # filtering the range [1, input] with
      !%QT        # lambda T: not (input % T)    (divisibility test)
                  # implicit print the
&                 # short-circuiting and of
 !%J        Q     # not (J % input)
             /J   # and J / input

Answer 10

JavaScript (ES6), 41 bytes

Returns 0 if there's no solution.

x=>(g=k=>x=k&&k*!(x%k)/x+g(k-1))(x)%1?0:x

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Answer 11

Jelly, 6 bytes

Æs0:%?

A monadic Link accepting a positive integer which yields a non-negative integer.

Try it online! Or see the test-suite.

How?

Æs0:%? - Link: x
Æs     - divisor sum
     ? - if...
    %  - ...condition: has a remainder when divided
  0    - ...then: zero
    :  - ...else: integeger divide

Answer 12

APL (Dyalog Unicode), 19 18 bytes

⊢(÷⍨×0=|)1⊥∘⍸0=⍳|⊢

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Conversion to train by Jo King.(-3 bytes)

-1 more byte from Jo King after changing the check condition.

Older answer, 22 bytes

{(⊢×⌊=⊢)⍵÷⍨+/⍸0=⍵|⍨⍳⍵}

Explanation

{(⊢×⌊=⊢)⍵÷⍨+/⍸0=⍵|⍨⍳⍵} ⍵ → input
                   ⍳⍵  range 1-⍵
                ⍵|⍨    mod ⍵
              0=       check which ones are divisors
             ⍸         get the indices (factors)
           +/          sum the factors
        ⍵÷⍨            divide by ⍵
 (⊢×⌊=⊢)               Inner tacit fn:
    ⌊=⊢                Floor equals right? (integer test, returns 0 or 1)
  ⊢×                   times right 

Answer 13

Haskell, 51 46 bytes

a!b=0^mod a b*div a b
f n=sum(map(n!)[1..n])!n

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Answer 14

Wolfram Language (Mathematica), 30 bytes

Tr@Divisors@#/#/._Rational->0&

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-6 bytes from @att

Answer 15

Charcoal, 23 20 bytes

ＮθＩ⌕Ｅ⊕θ×θιΣΦ⊕θ∧ι¬﹪θι

Try it online! Link is to verbose version of code. Port of @Sisyphus's algorithm but using @ovs's comment to deal with 0-indexing. Outputs -1 for nonexistence. Explanation:

Ｎθ                      Input `x` as a number
             θ          `x`
            ⊕           Incremented
           Φ            Filter over implicit range
               ι        Current index
              ∧         Logical AND
                  θ     `x`
                 ﹪      Modulo
                   ι    Current index
                ¬       Logical NOT
          Σ             Take the sum
      θ                 `x`
     ⊕                  Incremented
    Ｅ                   Map over implicit range
        θ               `x`
       ×                Multiplied by
         ι              Current index
   ⌕                    Find the index
  Ｉ                     Cast to string
                        Implicitly print

Unfortunately for Charcoal the sum of [] is not zero, which means that I can't save a byte by removing the two increments of x and incrementing the result instead.

Previous 23-byte solution:

Ｎθ≔ΣΦ⊕θ∧ι¬﹪θιη¿¬﹪ηθＩ÷ηθ

Try it online! Link is to verbose version of code. Explanation:

Ｎθ

Input x.

≔ΣΦ⊕θ∧ι¬﹪θιη

Create a list from 1..x, filter out numbers that don't divide x, and take the sum.

¿¬﹪ηθＩ÷ηθ

If x divides the sum then print the quotient.

Answer 16

Scala, 54 53 bytes

x=>{val s=1 to x filter(x%_<1)sum;s/x*(1-(s%x).sign)}

Try it in Scastie

Sums every divisor of x from 1 to x, inclusive. If that sum is divisible by x, it returns that divided by x, otherwise it returns 0.

Answer 17

Retina, 51 bytes

.+
*
|""Lw`^(.+)(?=\1*$)
^
$-1;
L$`^(.+);(\1)+$
$#2

Try it online! Link includes less slower test cases. Explanation:

.+
*

Convert the input to unary.

|""Lw`^(.+)(?=\1*$)

List all of the factors without delimiting them, thus summing them.

^
$-1;

Retrieve the original unary value.

L$`^(.+);(\1)+$
$#2

Count how many times it divides the sum. (Or output nothing if it does not.)

Answer 18

Octave, 36 34 bytes

@(x)~mod(s=~mod(x,r=1:x)*r',x)*s/x

Anonymous function that takes a floating-point or integer number as input. The last test case fails due to memory limitations.

Try it online! Or verify test cases.

Explanation

@(x)~mod(s=~mod(x,r=1:x)*r',x)*s/x

@(x)                                 % anonymous function with input x
                    1:x              % row vector [1 2 ... x]
                  r=                 % call that r
            mod(x,     )             % x modulo [1 2 ... x]. Gives a row vector
           ~                         % negate each element. Gives 1 for divisors
                         r'          % column vector [1; 2; ... ; x]
                        *            % matrix-multiply. Gives the sum of divisors
         s=                          % call that s
     mod(                  ,x)       % sum of divisors modulo x
    ~                                % negate. Gives 1 if x divides sum of divisors
                               s/x   % sum of divisors divided by x
                              *      % multiply

Answer 19

Python 3.8 (pre-release), 62 bytes

lambda x:(a:=sum(x/i*(x%i<1)for i in range(1,x+1)))%x<1and a/x

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Answer 20

MathGolf, 7 bytes

─Σk‼÷/*

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Explanation:

─       # Get the divisors of the (implicit) input-integer
 Σ      # Sum those divisors
  k     # Push the input-integer again
   ‼    # Apply the following two commands separately to the stack:
    ÷   #  Check if the divisor-sum is divisible by the input (1 if truthy; 0 if falsey)
    /   #  Integer-divide the divisor-sum by the input
     *  # Multiply the two together
        # (after which the entire stack joined together is output implicitly as result)

Answer 21

Rockstar, 141 135 131 bytes

Outputs nothing if no n exists.

listen to N
X's0
T's0
while N-X
let X be+1
let D be N/X
turn up D
let T be+D is N/X and X

let D be T/N
turn up D
if D is T/N
say D

Try it here (Code will need to be pasted in)

Answer 22

PowerShell, 71 bytes

param($x)(1..$x|?{!($x%$_)})|%{$s+=$_};((1..$x)|%{$_*$x}).indexof($s)+1

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• Get the parameter x
• Brute force and calculate divisors
• Sum up the divisors to variable s
• Multiply range of 1..x with x
• Return index of s in the multiplied range and add 1 to it

Answer 23

Icon, 67 bytes

procedure f(n)
s:=0
n%(i:=1to n)=0&s+:=i&\z
return(0=s%n&s/n)|0
end

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Answer 24

Japt -æ, 7 bytes

Outputs undefined if no n is found.

*N¶Îâ x

Try it

Answer 25

Factor, 84 bytes

: f ( n -- n ) dup [1,b] [ dupd mod 0 = ] filter sum swap /mod 0 > [ drop 0 ] when ;

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Answer 26

Prolog, 117 bytes

s(X,D,S):-D<1,!,S is 0;E is D-1,(0 is X mod D,!,s(X,E,T),S is T+D;s(X,E,S)).
f(X,N):-s(X,X,S),0 is S mod X,N is S//X.

Try it online! (Please don't modify it directly, it would change my version too)

---

If anyone could figure out why this shorter version (96 bytes) isn't working, I'd be really grateful.

s(X,D,S):-D<1,!,S is 0;E is D-1,(0 is X mod D,!,s(X,E,T),S is T+D;s(X,E,S)).
f(X,N):-s(X,X,N*X).

Version with print debugging

Answer 27

GolfScript, 22 bytes

~:x),{.x\%!*+}*.x%!*x/

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~:x                     # Store the input in x
   ),                   # Make an array from 0 to x
     {       }*         # For each number in the array, execute this block
      .                 # Copy current number
       x\%!             # The copy becomes 1 if it is a divisor of x and 0 if it isn't
           *+           # Multiply and add
               .        # Copy the sum of the divisors
                x%!     # The copy becomes 1 if it is a divisor of x and 0 if it isn't
                   *    # Multiply
                    x/  # Divide by x