# An abundance of integers!

An Abundant number is any number where the sum of its proper divisors is greater than the original number. For example, the proper divisors of 12 are:

1, 2, 3, 4, 6


And summing these results in 16. Since 16 is larger than 12, 12 is abundant. Note that this does not include "Perfect numbers", e.g. numbers that are equal to the sum of its proper divisors, such as 6 and 28.

Your task for today is to write a program or function that determines if a number is abundant or not. Your program should take a single integer as input, and output a truthy/falsy value depending on whether it is abundant or not. You can assume that the input will always be valid and greater than 0, so for bad inputs, undefined behavior is fine.

You may take your input and output in any reasonable format, for example STDIN/STDOUT, files, or arguments/return values would all be acceptable.

For reference, here are the abundant numbers up to 100:

12,
18,
20,
24,
30,
36,
40,
42,
48,
54,
56,
60,
66,
70,
72,
78,
80,
84,
88,
90,
96,
100


And more can be found on A005101

Since this is , standard loopholes are denied, and try to write the shortest possible code you can in whichever language you happen to choose!

• "the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!" – mbomb007 Feb 21 '17 at 14:29
• The asymptotic density of abundant numbers is somewhere within the interval [0.24761748, 0.24764422]. Calculated using the source code included in this paper. – Deadcode Jan 25 '19 at 22:20
• I am trying to do this in Geometry Dash. It's a nightmare – MilkyWay90 Jan 29 '19 at 3:25

# PowerShell, 43 bytes

param($n)1..$n|%{$s+=$_*!($n%$_)}
$s-gt2*$n


Try it online!

# Common Lisp, 63 bytes

(lambda(n)(>(loop for i from 1 below n if(=(mod n i)0)sum i)n))


Try it online!

## F#, 51 bytes

let f n=Seq.filter(fun x->n%x=0){1..n-1}|>Seq.sum>n


Try it online!

Filters out all the numbers that don't divide evenly into n, then sums them and compares them against n.

# Japt, 4 bytes

<âÔx


Try it online!

# C# (Visual C# Interactive Compiler), 48 bytes

x=>Enumerable.Range(1,x).Sum(y=>y<x&x%y<1?y:0)>x


Try it online!

Above is a LINQ based solution that creates a range of numbers slightly larger than the input number. Each element is tested for being a) being a proper divisor of the input and b) being less than the input. If both checks pass, it is added to a sum which is compared to the input.

Below is a a recursive solution that uses an inner function to calculate the summation. While this one seems the most interesting to me, I think the LINQ based approach will not be beat.

# C# (Visual C# Interactive Compiler), 60 bytes

x=>{int g(int y)=>(x%y<1?y:0)+(++y<x?g(y):0);return g(1)>x;}


Try it online!

Finally, we have an iterative solution which currently is between the LINQ based approach and the recursive one.

# C# (Visual C# Interactive Compiler), 53 bytes

x=>{int y=0,z=0;for(;++y<x;z+=x%y<1?y:0);return z>x;}


Try it online!

# APL (Dyalog Unicode), 11 bytesSBCS

⊢<1⊥∘⍸0=⍳|⊢


Try it online!

Explanation:

⊢<1⊥∘⍸0=⍳|⊢ ⍝ Monadic function train
⍳   ⍝ Generate integers from 1 to input
|⊢ ⍝ For each of the above, modulo the input
0=   ⍝ Return a boolean vector where ones correspond
⍝ to zeroes
⍸      ⍝ Return the indices of the ones in the above vector
⍝ This results in a vector of the proper divisors of our input
1⊥∘      ⍝ Composed with the above, decode into unary base.
⍝ This has the same effect as summing the vector, but is used
⍝ here to comply with the function train format.
⊢<         ⍝ Is the above sum greater than our input?


# R, 32 28 bytes

2*(n=scan())<(x=1:n)%*%!n%%x


Try it online!

Reads n from stdin, returns TRUE for abundant numbers and FALSE otherwise.

# Whispers v2, 49 bytes

> Input
>> ∤1
>> ∑2
>> 3-1
>> 4>1
>> Output 5


Try it online!

## How it works

Whispers is a non-linear language, where numbers don't refer to their numerical value; rather they reference the result of that line in the program. Execution always starts at the last line, and branches through these references.

Program structure tree (green are nilad lines): Therefore, we start on line 6 by outputting the result of line 5, which is

>> 4>1


which calls lines 4 and 1 and compares the results.

Line 1 is the odd line out in the program: as it starts with a single >, rather than a double >>, it is a nilad line. This means it returns its value interpreted literally. Numbers return their numeric value, and Input is preprocessed at the start of execution, being replaced with the first input on STDIN. Therefore, line 5 checks if the result for line 4 is greater than the input.

Line 4, >> 3-1, subtracts the input from line 3's result, while line 3, >>∑2, sums the values on line 2. Line 2, >> ∤1, generates the list of the input's divisors. Therefore, together, the program generates a list of the input's divisors (including the input), sums them, then subtracts the input (in order to account for the input being included in the list of divisors). Finally, we compare the sum to the input, and output the result.

# Zsh, 39 bytes

for ((;++b<$1;t+=$1%b?0:b)):
<<<$[t>$1]


Try it online!

# Lua (71 68 bytes)

Thank to @Deadcode for finding a typo in the byte length and for suggesting shorter code.

function A(n)z=0 for i=1,n-1 do z=z+i*#{[n%i+1]=0}end return z>n end

• Welcome to CCGC, and -2 bytes by using z+i*#{[n%i+1]=0} instead of n%i<1 and z+i or z - Try it online!. Also, how did you measure your length? It seems to be 54 bytes even without the function header and footer. – Deadcode Mar 2 at 3:36
• @Deadcode Typo fixed and suggestion included. Thank you for the nice trick with  – Jairo A. del Rio Mar 2 at 3:50

# x86 opcode edited from 179016, 20 bytes

f:      mov ecx, eax       ; copy input number N into ECX
lea ebx, [eax+eax] ; copy input number 2N into EBX
.a:     cdq                ; clear DX (high word for dividend)
push eax           ; save original dividend
div ecx            ; divide DX:AX / CX
pop eax            ; restore dividend
test edx, edx      ; if remainder = 0, it divides evenly and is a divisor of N
jnz .b             ; if not, continue loop
sub ebx, ecx       ; Subtract divisor from EBX; this simultaneously changes the
; running total, and sets the flags for the return value.
jc .c              ; If CF=1, the sum of divisors subtracted from N has exceeded
; N, thus N is abundant, and we need to exit the loop now to
; return this result in CF.
.b:     loop .a
.c:                        ; return value: CF=0 (non-abundant), CF=1 (abundant)


Try it online!

Saved from the reverse working

• Try it online! (please change to a direct link if you include it) Note that this TIO is not executing exactly the same machine code, as it needs to use 0x66 and 0x67 prefixes to do 16-bit operations, but I still think it's much better than nothing. :-) – Deadcode Mar 8 at 21:20
• @Deadcode Totally changed to 32-bit code and fixed N=1 case with the price that >=0x80000000 fail – l4m2 Mar 9 at 21:42
• 32-bit means you can use the hex-dump code I added: Try it online!. Cool that you chose a different way of dealing with input=1. In yours, values of 2^31 or larger can't be handled. In mine, all 32-bit inputs can be handled, making it 3 bytes longer than yours. (And of course, neither of them can handle input=0.) – Deadcode Mar 10 at 18:26

# ><>, 47+346+3 39+3 = 42 bytes

:l)?!v}l:{:}$%0=*{ v-$0<
l<+v?=1
)n;>0


Expects the input number to be present on the stack at program start, so +3 bytes for the -v flag. Try it online!

Edit: Golfed 1 byte from 3rd line: -\~0$ => -\:+ Edit 2: Near complete rewrite, previous version: 1v }/!?)@:{:}+1r~?$r}:}%@:{:
-\:+
?\+l1=
;>0)n


a n=n<sum[a|a<-[1..n-1],mod n a<1]


Try it online! Usage example: a 12 returns True.

# JavaScript ES6, 61 50 bytes

a=>[...Array(a).keys()].reduce((b,c)=>a%c?b:b+c)>a


f=a=>[...Array(a).keys()].reduce((b,c)=>a%c?b:b+c)>a;

console.log(f(11));
console.log(f(12));
console.log(f(17));
console.log(f(18));

• 45 bytes – Shaggy Sep 12 '17 at 15:51

# Pyth, 8 bytes

>s{*MPyP


Try it online here.

>s{*MPyPQQ   Implicit: Q=eval(input())
Trailing QQ inferred
PQ    Prime factors of Q
y      Take the powerset of the above
P       Discard the last one (i.e. the original set itself)
*M        Take the product of each list
{          Deduplicate
s           Take the sum
>        Q   Is the above greater than Q? Implicit print


# PHP, 63 bytes

function a($x){for($i=1;$i<$x;$i++){$s+=$x%$i?0:$i;}echo$s>$x;}  Usage: a(12); # APL(NARS) 11 chars, 22 bytes {⍵<2÷⍨11π⍵}  11π is the function sum divisor, so sum proper divisor would be: {(-⍵)+11π⍵}; test:  f←{⍵<2÷⍨11π⍵} {1=f ⍵:⍵⋄⍬}¨1..100 12 18 20 24 30 36 40 42 48 54 56 60 66 70 72 78 80 84 88 90 96 100  # Ruby, 39 32 bytes ->n{n<(1...n).sum{|x|n%x>0?0:x}}  Try it online! Updated to Ruby 2.6 # Forth (gforth), 45 bytes : f 0 over 1 ?do over i mod 0= i * - loop < ;  Try it online! ### Explanation Loops over every number from 1 to n-1, summing all values that divide n perfectly. Returns true if sum is greater than n ### Code Explanation : f \ start word definition 0 over 1 \ create a value to hold the sum and setup up the bounds of the loop ?do \ start a counted loop from 1 to n. (?do skips if start = end) over \ copy n to the top of the stack i mod 0= \ check if i divides n perfectly i * - \ if so, use the fact that -1 = true in forth to add i to the sum loop \ end the counted loop < \ check if n is less than the sum ; \ end the word definition  # C# (.NET Core), 59, 53 bytes Without LINQ. EDIT: Thanks to ASCII-only for pointing out for(;++k<p;) instead of for(;k<p;k++). # C# (.NET Core), 53 bytes p=>{int k=0,n=0;for(;++k<p;)n+=p%k<1?k:0;return n>p;}  Try it online! • 53 – ASCII-only Mar 13 '19 at 5:56 # Perl 5-MList::Util=sum -pa, 32 bytes $_=$_<sum grep!("@F"%$_),1..\$_-1


Try it online!

# Scala, 39 bytes

(n:Int)=>(1 until n filter{n%_<1}sum)>n


# Javascript, 47 Bytes

a=>{s=d=0;while(a>d){s+=a%d?0:d;d++}return s>d}

f=a=>{s=d=0;while(a>d){s+=a%d?0:d;d++}return s>d};
console.log(f(11));
console.log(f(12));
console.log(f(15));
console.log(f(18));

Quite straight forward: check all numbers smaller than the input, if the input is divisible by them add them to the sum, and return whether the sum is bigger than the input

# C, 68 bytes

int s,x;main(i){scanf("%d",&x);while(++i<x)s+=x%i^0?0:i;return s>x;}

• Welcome to the site! I think you could save some bytes by taking input from function arguments, rather than implementing main with scanf. For example int f(int s,int x){while(++i<x)s+=x%i^0?0:i;return s>x;} Also, I'm not an expert at C golfing, but I think some compilers default to int functions with int arguments, so you could do f(s,x){while(++i<x)s+=x%i^0?0:i;return s>x;}, but you'd have to find the right compiler and test it first. – James Feb 21 '17 at 17:59

# Java 7, 69 68 bytes

Object c(int n){int s=0,i=1;for(;++i<=n/2;s+=n%i<1?i:0);return s>n;}


Explanation:

Object c(int n){                       // Boolean method with integer parameter
int s = 0,                          // sum initialization
i = 1;                          // counter initialization (starts at 2: i=1 and ++i becomes i=2)
for(; ++i <= n/2; s += n % i < 1    // loop while (i <= n/2)
? i         //   if (n % i == 0)
: 0);       //     r = r + i
return s > n;                       // return sum > input
}


Test code:

Try it here.

class M{
static Object c(int n){int s=0,i=1;for(;++i<=n/2;s+=n%i<1?i:0);return s>n;}

public static void main(String[] a){
for(int i = 0; i <= 100; i++){
System.out.print(i+": "+c(i) + ";  ");
}
}
}


Output:

0: false;  1: false;  2: false;  3: false;  4: false;  5: false;  6: false;  7: false;  8: false;  9: false;  10: false;
11: false;  12: true;  13: false;  14: false;  15: false;  16: false;  17: false;  18: true;  19: false;
20: true;  21: false;  22: false;  23: false;  24: true;  25: false;  26: false;  27: false;  28: false;  29: false;
30: true;  31: false;  32: false;  33: false;  34: false;  35: false;  36: true;  37: false;  38: false;  39: false;
40: true;  41: false;  42: true;  43: false;  44: false;  45: false;  46: false;  47: false;  48: true;  49: false;
50: false;  51: false;  52: false;  53: false;  54: true;  55: false;  56: true;  57: false;  58: false;  59: false;
60: true;  61: false;  62: false;  63: false;  64: false;  65: false;  66: true;  67: false;  68: false;  69: false;
70: true;  71: false;  72: true;  73: false;  74: false;  75: false;  76: false;  77: false;  78: true;  79: false;
80: true;  81: false;  82: false;  83: false;  84: true;  85: false;  86: false;  87: false;  88: true;  89: false;
90: true;  91: false;  92: false;  93: false;  94: false;  95: false;  96: true;  97: false;  98: false;  99: false;  100: true;

• Sorry, I was looking at the main code in that part but the golfed code above. Reading failure. – user18932 Feb 22 '17 at 2:45
• I am most likely overlooking something obvious, though I am curious: Why did you choose to use Object as opposed to bool? – Jonathan Frech Jan 22 '19 at 23:14
• @JonathanFrech In Java it's boolean, so it's 1 byte longer than Object :) Although nowadays I would just use a Java 8+ lambda, so Object c(int n) would be n->. – Kevin Cruijssen Jan 23 '19 at 7:46

# Racket, 46 bytes

That's 46 bytes including the (require math).

(λ(n)(>(-(sum(divisors n))n)n))


The function produces booleans.

Try it online!

It's pretty simple, uses the divisors function to produce a list of all divisors of n. Since the list includes n itself, we need to remove it after summing.

• Hello and welcome to PPCG. Nice first post; however, I would think that you ought to include the byte count for your (require math) import. – Jonathan Frech Jan 23 '19 at 21:45
• Sorry! I was unsure about what exactly to add to the byte count. – waf9000 Jan 23 '19 at 22:20
• No problem. Usually, one counts everything the function or program truly needs -- excluding boiler code only written for testing. However, on edge cases one sometimes has to resort to the current consensus -- written and talked about on meta. In practice, you will often be notified by comment if someone disagrees with your byte count. – Jonathan Frech Jan 23 '19 at 22:25

# Java 8, 66 bytes

n->java.util.stream.IntStream.range(1,n).filter(i->n%i==0).sum()>n


# Explanation

n->                                       // define a lambda function
java.util.stream.IntStream.range(1,n) // create a stream of integers from 1 to n
.filter(i->n%i==0)                    // filter out non-factors
.sum()                                // sum the factors
>n                                        // compare the sum to the original number n
// implicit return


# Javascript,72bytes

var n=prompt(),div=0,i=1;while(++i<=n/2){div+=n%i?0:i}console.log(div>n)

• As @SriotchilismO'Zaic points out, this appears to be a snippet which is not compliant with our current I/O consensus. Please fix your post accordingly. – Jonathan Frech Mar 13 '19 at 8:32
• As @JonathanFrech specified, modified the changes and further refined the code. – Jeyanth Mar 13 '19 at 9:58

# Kotlin, 37 bytes

{i->(2..i-1).filter{i%it==0}.sum()>i}


Expanded:

{ i -> (2..i - 1).filter { i % it == 0 }.sum() > i }


Explanation:

  {i->                      start lambda with parameter i
(2..i-1)                create a range from 2 until i - 1 (potential divisors)
.filter{i%it==0}      it's a divisor if remainder = 0; filter divisors
>i                compare to the original number


Try it online!

# TI-BASIC (TI-84), 23 bytes

:2Ans<sum(seq(Ans/Xnot(remainder(Ans,X)),X,1,Ans,1


Input is in Ans.
Output is in Ans and is automatically printed out when the program completes.

Explanation:
(TI-BASIC doesn't have comments, so just assume that ; makes a comment)

:2Ans=sum(seq(Ans/Xnot(remainder(Ans,X)),X,1,Ans    ;Full program

2Ans                                               ;double the input
seq(                                      ;generate a list
X,          ;using the variable X,
1,        ;starting at 1,
Ans     ;and ending at the input
;with an implied increment of 1
Ans/X                                 ;from the input divided by X
not(                ),           ;multiplied by the negated result of
remainder(Ans,X)              ;the input modulo X
;(result: 0 or 1)
sum(                                          ;sum up the elements in the list
<                                              ;is the sum greater than double the input?


Example:

12
12
prgmCDGF4
1
6
6
prgmCDGF4
0


Note: The byte count of a program is evaluated using the value in [MEM]>> (36 bytes) then subtracting the length of the program's name, CDGF2, (5 bytes) and an extra 8 bytes used for storing the program:

36 - 5 - 8 = 23 bytes

• I counted 50 characters, so presumably 50 bytes... what am I missing? – Adam Mar 13 '19 at 14:33
• @Adam all of the functions in TI-BASIC are 1-byte tokens. Therefore, sum(, seq(, remainder(, and not( would all be considered as 1 byte each. Ans is also a 1-byte token. – absoluteAquarian Mar 13 '19 at 14:40
• Ah, okay. I see now – Adam Mar 13 '19 at 14:42
• Actually, I should clarify that the remainder( function is a 2-byte token, not 1-byte. – absoluteAquarian Mar 13 '19 at 16:23