Sum of all divisors from 1 to n

Given a positive integer n. Find the value of

$\sum_{i=1}^{i=n} F(i)$

where function F(i) for number i be defined as the sum of all divisors of ‘i‘.

Examples :

Input: 4
Output: 15
Explanation
F(1) = 1
F(2) = 1 + 2 = 3
F(3) = 1 + 3 = 4
F(4) = 1 + 2 + 4 = 7
ans = F(1) + F(2) + F(3) + F(4)
    = 1 + 3 + 4 + 7
    = 15
Input: 5
Output: 21

Naive approach is to traverse for every number(1 to n), find all divisors and keep updating the sum with that divisor. See this to understand more.

C++

// C++ program to find sum of all
// divisor of number up to 'n'
#include<bits/stdc++.h>
using namespace std;
 
// Utility function to find sum of
// all divisor of number up to 'n'
int divisorSum(int n)
{
    int sum = 0;
 
    for(int i = 1; i <= n; ++i)
    {
         
        // Find all divisors of i and add them
        for(int j = 1; j * j <= i; ++j)
        {
            if (i % j == 0)
            {
                if (i / j == j)
                    sum += j;
                else
                    sum += j + i / j;
            }
        }
    }
    return sum;
}
 
// Driver code
int main()
{
    int n = 4;
    cout << " " << divisorSum(n) << endl;
     
    n = 5;
    cout << " " << divisorSum(n);
     
    return 0;
}

Java

// JAVA program to find sum of all
// divisor of number up to 'n'
import java.io.*;
 
class GFG {
 
    // Utility function to find sum of
    // all divisor of number up to 'n'
    static int divisorSum(int n)
    {
        int sum = 0;
 
        for (int i = 1; i <= n; ++i) {
 
            // Find all divisors of i
            // and add them
            for (int j = 1; j * j <= i; ++j) {
                if (i % j == 0) {
                    if (i / j == j)
                        sum += j;
                    else
                        sum += j + i / j;
                }
            }
        }
        return sum;
    }
 
    // Driver code
    public static void main(String args[])
    {
        int n = 4;
        System.out.println(divisorSum(n));
        n = 5;
        System.out.println(divisorSum(n));
    }
}
 
/*This code is contributed by Nikita tiwari.*/

Python3

# Python3 code to find sum of all
# divisor of number up to 'n'
 
# Utility function to find sum of
# all divisor of number up to 'n'
def divisorSum( n ):
    sum = 0
     
    for i in range(1, n + 1):
         
        # Find all divisors of i
        # and add them
        j = 1
        while j * j <= i:
            if i % j == 0:
                if i / j == j:
                    sum += j
                else:
                    sum += j + i / j
            j = j + 1
    return int(sum)
 
# Driver code
n = 4
print( divisorSum(n))
n = 5
print( divisorSum(n))
 
# This code is contributed by "Sharad_Bhardwaj".

C#

// C# program to find sum of all
// divisor of number up to 'n'
using System;
 
class GFG {
 
    // Utility function to find sum of
    // all divisor of number up to 'n'
    static int divisorSum(int n)
    {
        int sum = 0;
 
        for (int i = 1; i <= n; ++i) {
 
            // Find all divisors of i
            // and add them
            for (int j = 1; j * j <= i; ++j) {
                if (i % j == 0) {
                    if (i / j == j)
                        sum += j;
                    else
                        sum += j + i / j;
                }
            }
        }
        return sum;
    }
 
    // Driver code
    public static void Main()
    {
        int n = 4;
        Console.WriteLine(divisorSum(n));
        n = 5;
        Console.WriteLine(divisorSum(n));
    }
}
 
/*This code is contributed by vt_m.*/

Javascript

<script>
// Javascript program to find sum of all
// divisor of number up to 'n'
 
// Utility function to find sum of
// all divisor of number up to 'n'
 
function divisorSum(n)
{
    let sum = 0;
 
    for (let i = 1; i <= n; ++i)
    {
 
        // Find all divisors of i
        // and add them
        for (let j = 1; j * j <= i; ++j)
        {
            if (i % j == 0)
            {
                if (i / j == j)
                    sum += j;
                else
                    sum += j + i / j;
            }
        }
    }
    return sum;
}
 
// Driver code
let n = 4;
document.write(divisorSum(n) + "<br>");
n = 5;
document.write(divisorSum(n) + "<br>");
 
// This code is contributed by _saurabh_jaiswal
</script>

PHP

<?php
// PHP program to find sum of all
// divisor of number up to 'n'
 
// Utility function to find sum of
// all divisor of number up to 'n'
 
function divisorSum($n)
{
    $sum = 0;
 
    for ($i = 1; $i <= $n; ++$i)
    {
 
        // Find all divisors of i
        // and add them
        for ($j = 1; $j * $j <= $i; ++$j)
        {
            if ($i % $j == 0)
            {
                if ($i / $j == $j)
                    $sum += $j;
                else
                    $sum += $j + $i / $j;
            }
        }
    }
    return $sum;
}
 
// Driver code
$n = 4;
echo "\n", divisorSum($n), "\n";
$n = 5;
echo divisorSum($n), "\n";
 
// This code is contributed by aj_36
?>

Output

 15
 21

Time complexity: O(n√(n)))
Auxiliary space: O(1)

Efficient approach is to observe the function and co-relate the pattern. For a given number n, every number from 1 to n contributes its presence up to the highest multiple less than n. For instance,

Let n = 6,
=> F(1) + F(2) + F(3) + F(4) + F(5) + F(6)
=> 1 will occurs 6 times in F(1), F(2),
   F(3), F(4), F(5) and F(6)
=> 2 will occurs 3 times in F(2), F(4) and
   F(6)
=> 3 will occur 2 times in F(3) and F(6)
=> 4 will occur 1 times in F(4)
=> 5 will occur 1 times in F(5)
=> 6 will occur 1 times in F(6)

From the above observation, it can easily be observed that number i is occurring only in their multiples less than or equal to n. Thus, we just need to find the count of multiples and then multiply it with i for full contribution in the final sum. It can easily be done in O(1) time by taking the floor of (n / i) and then multiply it with i for the sum.

C++

// C++ program to find sum of all
// divisor of number up to 'n'
#include<bits/stdc++.h>
using namespace std;
 
// Utility function to find sum of
// all divisor of number up to 'n'
int divisorSum(int n)
{
    int sum = 0;
    for (int i = 1; i <= n; ++i)
        sum += (n / i) * i;
    return sum;
}
 
// Driver code
int main()
{
    int n = 4;
    cout <<" "<< divisorSum(n)<<endl;
    n = 5;
    cout <<" "<< divisorSum(n)<< endl;
    return 0;
}
 
// This code is contributed by shivanisinghss2110

C

// C program to find sum of all
// divisor of number up to 'n'
#include <stdio.h>
 
// Utility function to find sum of
// all divisor of number up to 'n'
int divisorSum(int n)
{
    int sum = 0;
    for (int i = 1; i <= n; ++i)
        sum += (n / i) * i;
    return sum;
}
 
// Driver code
int main()
{
    int n = 4;
    printf("%d\n", divisorSum(n));
    n = 5;
    printf("%d", divisorSum(n));
    return 0;
}

Java

// Java program to find sum of all
// divisor of number up to 'n'
import java.io.*;
 
class GFG {
 
    // Utility function to find sum of
    // all divisor of number up to 'n'
    static int divisorSum(int n)
    {
        int sum = 0;
        for (int i = 1; i <= n; ++i)
            sum += (n / i) * i;
        return sum;
    }
 
    // Driver code
    public static void main(String args[])
    {
        int n = 4;
        System.out.println(divisorSum(n));
        n = 5;
        System.out.println(divisorSum(n));
    }
}
 
/*This code is contributed by Nikita Tiwari.*/

Python3

# Python3 code to find sum of all
# divisor of number up to 'n'
 
# Utility function to find sum of
# all divisor of number up to 'n'
def divisorSum( n ):
    sum = 0
    for i in range(1, n + 1):
        sum += int(n / i) * i
    return int(sum)
     
# Driver code
n = 4
print( divisorSum(n))
n = 5
print( divisorSum(n))
 
# This code is contributed by "Sharad_Bhardwaj".

C#

// C# program to find sum of all
// divisor of number up to 'n'
using System;
 
class GFG {
 
    // Utility function to find sum of
    // all divisor of number up to 'n'
    static int divisorSum(int n)
    {
        int sum = 0;
        for (int i = 1; i <= n; ++i)
            sum += (n / i) * i;
        return sum;
    }
 
    // Driver code
    public static void Main()
    {
        int n = 4;
        Console.WriteLine(divisorSum(n));
        n = 5;
        Console.WriteLine(divisorSum(n));
    }
}
 
/*This code is contributed by vt_m.*/

Javascript

// Javascript program to find sum of all
// divisor of number up to 'n'
 
// Utility function to find sum of
// all divisor of number up to 'n'
function divisorSum(n)
{
    let sum = 0;
    for (let i = 1; i <= n; ++i)
        sum += Math.floor(n / i) * i;
    return sum;
}
 
// Driver code
let n = 4;
document.write(divisorSum(n) + "<br>");
n = 5;
document.write(divisorSum(n) + "<br>");
 
// This code is contributed by _saurabh_jaiswal.

PHP

<?php
// PHP program to find sum of all
// divisor of number up to 'n'
 
// Utility function to find sum of
// all divisor of number up to 'n'
function divisorSum( $n)
{
    $sum = 0;
    for ( $i = 1; $i <= $n; ++$i)
        $sum += floor($n / $i) * $i;
    return $sum;
}
 
// Driver code
$n = 4;
echo divisorSum($n),"\n";
$n = 5;
echo divisorSum($n),"\n";
 
// This code is contributed by anuj_67.
?>

Output

 15
 21

Time complexity: O(n)
Auxiliary space: O(1)

More efficient solution:

We need to calculate

$\sum_{i=1}^{i=N} (i*\lfloor N/i \rfloor )$

To evaluate the above expression in O(sqrt(N)) we make use of The Harmonic Lemma.

Consider the harmonic sequence on integer division: {N/1, N/2, N/3, ….. ,N/N}

The lemma states that the above sequence is non-increasing, and there are at most 2*sqrt(N) different elements.

Consider floor(N/i) = k. Thus, k <= N/i < k+1. From this we get largest = floor(N/k). Therefore, we can find a range of values of i for which floor(N/i) is constant. And using The Harmonic Lemma we know that will be at most 2*sqrt(N) terms, thus we can calculate it programmatically in O(sqrt(N)) complexity. Consider the following example for better clarification.

C++

// C++ program to calculate sum of divisors
// of numbers from 1 to N in O(sqrt(N)) complexity
#include <iostream>
using namespace std;
 
#define ll long long
#define mod 1000000007
 
/*
Function to calculate x^y using
Modular exponentiation
Refer to https://www.geeksforgeeks.org/
modular-exponentiation-power-in-modular-arithmetic/
*/
ll power(ll x, ll y, ll p)
{
     
    // re x^y if p not specified
    // else (x^y)%p
    ll res = 1;
    x = x % p;
    while (y > 0)
    {
        if (y & 1)
            res = (res * x) % p;
        y = y >> 1;
        x = (x * x) % p;
    }
    return (res + p) % p;
}
 
// Function to find modular
// inverse of a under modulo m
// Assumption: m is prime
ll modinv(ll x)
{
    return power(x, mod - 2, mod);
}
 
// Function to calculate sum from 1 to n
ll sum(ll n)
{
    // sum 1 to n = (n*(n+1))/2
    ll retval = ((((n % mod) * ((n + 1) %
        mod)) % mod) * modinv(2)) % mod;
    return retval;
}
 
ll divisorSum(ll n)
{
    ll l = 1;
    ll ans = 0;
 
    while (l <= n)
    {
        ll k = n / l;
        ll r = n / k;
        k %= mod;
         
        // For i=l to i=r, floor(n/i) will be k
        ans += ((sum(r) - sum(l - 1) %
                        mod) * k) % mod;
         
        // Since values can be very large
        // we need to take mod at every step
        ans %= mod;
        l = r + 1;
    }
    ans = ans % mod;
      // ans can be negative
      // for example n = 831367 ans would be -534577982
    if (ans < 0){
        return ans+mod;
    }else{
        return ans;
    }
}
 
/* Driver program to test above function */
int main()
{
    int n = 5;
    cout << "The sum of divisors of all  numbers from 1 to " << n << " is: " \
                            << divisorSum(n) << '\n';
 
    n = 14;
    cout << "The sum of divisors of all numbers from 1 to " << n << " is: " \
                            << divisorSum(n) << '\n';
}

Java

// Java program to calculate
// sum of divisors of numbers
// from 1 to N in O(sqrt(N))
// complexity
import java.util.*;
class Main{
     
static int mod = 1000000007;
     
/*
Function to calculate x^y using 
Modular exponentiation
Refer to https://www.geeksforgeeks.org/
modular-exponentiation-power-in-
modular-arithmetic/
*/
public static long power(long x,
                         long y,
                         long p)
{
  // re x^y if p not specified 
  // else (x^y)%p
  long res = 1;
  x = x % p;
   
  while (y > 0)
  {
    if ((y & 1) != 0)
      res = (res * x) % p;
    y = y >> 1;
    x = (x * x) % p;
  }
  return (res + p) % p;
}
 
// Function to find modular 
// inverse of a under modulo m
// Assumption: m is prime
public static long modinv(long x)
{
  return power(x, mod - 2, mod);
}
 
// Function to calculate sum
// from 1 to n
public static long sum(long n)
{
  // sum 1 to n = (n*(n+1))/2
  long retval = ((((n % mod) * ((n + 1) %
                    mod)) % mod) * modinv(2)) %
                    mod;
  return retval;
}
       
public static long divisorSum(long n)
{
  long l = 1;
  long ans = 0;
 
  while (l <= n)
  {
    long k = n / l;
    long r = n / k;
    k %= mod;
 
    // For i=l to i=r,
    // floor(n/i) will be k
    ans += ((sum(r) - sum(l - 1) %
             mod) * k) % mod;
 
    // Since values can be very
    // large we need to take mod
    // at every step
    ans %= mod;
    l = r + 1;
  }
  ans = ans % mod;
  return ans;
}
 
// Driver code   
public static void main(String[] args)
{
  int n = 5;
  System.out.println("The sum of divisors of" +
                     " all numbers from 1 to " +
                     n + " is: " + divisorSum(n));
 
  n = 14;
  System.out.println("The sum of divisors of all" +
                     " numbers from 1 to " + n +
                     " is: " + divisorSum(n));
}
}
 
// This code is contributed by divyeshrabadiya07

Python3

# Python program to calculate
# sum of divisors of numbers
# from 1 to N in O(sqrt(N))
# complexity
mod = 1000000007;
 
# Function to calculate x^y using Modular exponentiation Refer to
# https:#www.geeksforgeeks.org/ modular-exponentiation-power-in-
# modular-arithmetic/
def power(x, y, p):
     
    # re x^y if p not specified
    # else (x^y)%p
    res = 1;
    x = x % p;
 
    while (y > 0):
        if ((y & 1) != 0):
            res = (res * x) % p;
        y = y >> 1;
        x = (x * x) % p;
     
    return (res + p) % p;
 
# Function to find modular
# inverse of a under modulo m
# Assumption: m is prime
def modinv(x):
    return power(x, mod - 2, mod);
 
# Function to calculate sum
# from 1 to n
def sum(n):
   
    # sum 1 to n = (n*(n+1))/2
    retval = ((((n % mod) * ((n + 1) % mod)) % mod) * modinv(2)) % mod;
    return retval;
 
def divisorSum(n):
    l = 1;
    ans = 0;
 
    while (l <= n):
        k = n // l;
        r = n // k;
        k %= mod;
 
        # For i=l to i=r,
        # floor(n/i) will be k
        ans += ((sum(r) - sum(l - 1) % mod) * k) % mod;
 
        # Since values can be very
        # large we need to take mod
        # at every step
        ans %= mod;
        l = r + 1;
     
    ans = ans % mod;
    return ans;
 
# Driver code
if __name__ == '__main__':
    n = 5;
    print("The sum of divisors of all numbers from 1 to " , n , " is: " ,int( divisorSum(n)));
 
    n = 14;
    print("The sum of divisors of all numbers from 1 to ", n ," is: " , int(divisorSum(n)));
 
# This code contributed by aashish1995 Write

C#

// C# program to calculate
// sum of divisors of numbers
// from 1 to N in O(sqrt(N))
// complexity
using System;
 
class GFG{
     
static int mod = 1000000007;
 
/*
Function to calculate x^y using 
Modular exponentiation
Refer to https://www.geeksforgeeks.org/
modular-exponentiation-power-in-
modular-arithmetic/
*/
static long power(long x, long y, long p)
{
     
    // re x^y if p not specified 
    // else (x^y)%p
    long res = 1;
    x = x % p;
     
    while (y > 0)
    {
        if ((y & 1) != 0)
            res = (res * x) % p;
             
        y = y >> 1;
        x = (x * x) % p;
    }
    return (res + p) % p;
}
 
// Function to find modular 
// inverse of a under modulo m
// Assumption: m is prime
static long modinv(long x)
{
    return power(x, mod - 2, mod);
}
 
// Function to calculate sum
// from 1 to n
static long sum(long n)
{
     
    // sum 1 to n = (n*(n+1))/2
    long retval = ((((n % mod) * ((n + 1) %
                  mod)) % mod) * modinv(2)) %
                  mod;
    return retval;
}
    
static long divisorSum(long n)
{
    long l = 1;
    long ans = 0;
     
    while (l <= n)
    {
        long k = n / l;
        long r = n / k;
        k %= mod;
         
        // For i=l to i=r,
        // floor(n/i) will be k
        ans += ((sum(r) - sum(l - 1) %
                   mod) * k) % mod;
         
        // Since values can be very
        // large we need to take mod
        // at every step
        ans %= mod;
        l = r + 1;
    }
    ans = ans % mod;
    return ans;
}
 
// Driver code
static void Main()
{
    int n = 5;
    Console.WriteLine("The sum of divisors of" +
                      " all numbers from 1 to " +
                      n + " is: " + divisorSum(n));
     
    n = 14;
    Console.WriteLine("The sum of divisors of all" +
                      " numbers from 1 to " + n +
                      " is: " + divisorSum(n));
}
}
 
// This code is contributed by divyesh072019

Javascript

<script>
 
// Javascript program to calculate
// sum of divisors of numbers
// from 1 to N in O(sqrt(N))
// complexity
 
var mod = 10007;
     
/*
Function to calculate x^y using 
Modular exponentiation
Refer to https://www.geeksforgeeks.org/
modular-exponentiation-power-in-
modular-arithmetic/
*/
function power(x, y, p)
{
  // re x^y if p not specified 
  // else (x^y)%p
  var res = 1;
  x = x % p;
   
  while (y > 0)
  {
    if ((y & 1) != 0)
      res = (res * x) % p;
    y = y >> 1;
    x = (x * x) % p;
  }
  return (res + p) % p;
}
 
// Function to find modular 
// inverse of a under modulo m
// Assumption: m is prime
function modinv(x)
{
  return power(x, mod - 2, mod);
}
 
// Function to calculate sum
// from 1 to n
function sum(n)
{
  // sum 1 to n = (n*(n+1))/2
  var retval = Math.floor((((n % mod) * ((n + 1) %
                    mod)) % mod) * modinv(2)) %
                    mod;
  return retval;
}
       
function divisorSum(n)
{
  var l = 1;
  var ans = 0;
 
  while (l <= n)
  {
    var k = n / l;
    var r = n / k;
    k %= mod;
 
    // For i=l to i=r,
    // floor(n/i) will be k
    ans += Math.floor((sum(r) - sum(l - 1) %
             mod) * k) % mod;
 
    // Since values can be very
    // large we need to take mod
    // at every step
    ans %= mod;
    l = r + 1;
  }
  ans = ans % mod;
  return ans;
}
 
// Driver code   
  var n = 5;
  document.write("The sum of divisors of" +
                     " all numbers from 1 to " +
                     n + " is: " + divisorSum(n) +"<br>");
 
  n = 14;
  document.write("The sum of divisors of all" +
                     " numbers from 1 to " + n +
                     " is: " + divisorSum(n));
 
// This code is contributed by shivanisinghss2110
</script>

Output

The sum of divisors of all  numbers from 1 to 5 is: 21
The sum of divisors of all numbers from 1 to 14 is: 165

Time complexity: O(sqrt(N))

Auxiliary space: O(1)

Another sqrt(n) approach:

Anywhere division is used in the below article, it means integer division.

Let’s start with an example assume that n = 20, now let’s see how each number from 1 to 20 appears as the factor of some other number.

 1 : 1 * 1, 1 * 2, 1 * 3, 1 * 4..., 1 * (20 / 1)
 2 : 2 * 1, 2 * 2, 2 * 3, 2 * 4,..., 2 * (20 / 2)
 3 : 3 * 1, 3 * 2, 3 * 3, 3 * 4...., 3 * (20 / 3)

our goal is to add every number each time it appears as the factor of some other number. For example 3 appears as the factor of (3 * 1), (3 * 2), (3 * 3)…, (3 * (20 / 3)). Now let’s start from 1 and add 1 to our sum each time it appears and also we will add all the numbers that appeared with 1, we’ll do the same thing with 2 and when we reach 3 we have already added 3 to our sum when it appeared with 1 and 2 so now we will only add 3 when it appears with numbers greater than 2 i.e. 3, 4, 5, 6 also we will add the numbers that appeared with 3 so we’ll add 4, 5 and 6 as well (notice here we will not add 3 twice because of 3 * 3). Similarly when we reach 4 we have already added 4 when it appeared with 1, 2 and 3 so we’ll add it only when it appears with numbers >= itself and add the numbers that appear with 4.

Finally we can say that when we are at a number i, we have already processed numbers from1 to i – 1 and hence we have added i every time it appears with numbers 1 to i – 1 so this time we only need to add i every time it appears with numbers >= i also we have to add all the numbers that appear together with i and they are > i.

Therefore for every number i we want to add the following terms to our sum

t1 : (add i each time it appears with numbers >= itself) -> i * (num / i - (i - 1)) 
(recall i will appear with numbers 1 to num / i 
and we have already added i each time it appeared with a numbers less than itself)
t2 : (add numbers that appear with i) -> (i + 1) + (i + 2) ... + (num / i) 
(numbers 1 to num / i will appear with i but 
we have already processed numbers 1 to i - 1 and added them 
when they appeared with i so now we only have to add the numbers 
that appear with i and are greater than i, 
here we will not add i itself because when i appears with itself 
it should be added only once and we have added it once in t1)
we need to calculate t2 in O(1) time, here's how to do that
t2 = (i + 1) + (i + 2) + (i + 3) + ... + (num / i)
add and subtract 1 + 2 + 3 ... + i
=> t2 = 1 + 2 + 3 + ... + i + (i + 1) + (i + 2) + ... + (num / i) - (1 + 2 + 3 + ... + i)
=> t2 = (1 + 2 + 3 + .. + (num / i)) - (1 + 2 + 3 .. + i)
=> t2 = ((num / i) * (num / i + 1)) / 2 - (i * (i + 1)) / 2

Finally, let’s look at the numbers that are greater than sqrt(num). These numbers will only appear with numbers that are lesser than sqrt(num). Let’s say x is a number greater than sqrt(num)

we have,
x > sqrt(num)
multiply sqrt(num) on both sides
=> x * sqrt(num) > sqrt(num) * sqrt(num)
=> x * sqrt(num) > num

we want to add x each time it appears, from above proof we see that x multiplied by root of num itself is greater than num hence x will only appear with numbers less than root of num so if we process all the numbers from 1 to sqrt(num) we will add every time x appears. For example take n = 100 now consider 11, 11 * 10 > 100 so 11 appears only with 1 to 9 i.e. as a factor of 11, 22, 33,…, 99 same is true for rest of the numbers that are greater than 10 they will only appear with numbers lesser than 10 and hence we only need to process numbers from 1 to 10 to add the numbers greater than 10 for n = 100.

Finally, our solution is this

for each i in 1 to sqrt(num) //no need to visit numbers greater than the root
    add t1 and t2 to the sum

below is the c++ code

C++

#include <bits/stdc++.h>
using namespace std;
long long sum_all_divisors(long long num)
{
    long long sum = 0;
    for (long long i = 1; i <= sqrt(num); i++) {
        long long t1 = i * (num / i - i + 1); // adding i every time it appears with numbers greater than or equal to itself
        long long t2 = (((num / i) * (num / i + 1)) / 2) - ((i * (i + 1)) / 2); // adding numbers that appear with i and are greater than i
        sum += t1 + t2;
    }
    return sum;
}
int main()
{
    int n;
    long long sum = sum_all_divisors(n);
    cout << sum << '\n';
    return 0;
}

Java

import java.io.*;
 
class GFG {
 
public static int sum_all_divisors(int num)
{
    int sum = 0;
    for (int i = 1; i <= Math.sqrt(num); i++) {
        int t1 = i * (num / i - i + 1); // adding i every time it appears with numbers greater than or equal to itself
        int t2 = (((num / i) * (num / i + 1)) / 2) - ((i * (i + 1)) / 2); // adding numbers that appear with i and are greater than i
        sum += t1 + t2;
    }
    return sum;
}
   
// Driver code 
public static void main (String[] args)
{
    int n = 1;
    int sum = sum_all_divisors(n);
    System.out.println(sum);
}
}
 
// This code is contributed by shivanisinghss2110

Python3

import math
def sum_all_divisors(num):
 
    sum = 0;
    for i in range(1,math.floor(math.sqrt(num))+1):
        t1 = i * (num / i - i + 1) # adding i every time it appears with numbers greater than or equal to itself
        t2 = (((num / i) * (num / i + 1)) / 2) - ((i * (i + 1)) / 2) # adding numbers that appear with i and are greater than i
        sum += t1 + t2;
     
    return sum;
 
n = 1
sum = sum_all_divisors(n)
print(sum)
 
# This code is contributed by shivanisinghss2110

C#

using System;
 
class GFG {
 
public static int sum_all_divisors(int num)
{
    int sum = 0;
    for (int i = 1; i <= Math.Sqrt(num); i++) {
        int t1 = i * (num / i - i + 1); // adding i every time it appears with numbers greater than or equal to itself
        int t2 = (((num / i) * (num / i + 1)) / 2) - ((i * (i + 1)) / 2); // adding numbers that appear with i and are greater than i
        sum += t1 + t2;
    }
    return sum;
}
   
// Driver code 
public static void Main (String[] args)
{
    int n = 1;
    int sum = sum_all_divisors(n);
    Console.Write(sum);
}
}
 
// This code is contributed by shivanisinghss2110

Javascript

<script>
function sum_all_divisors(num)
{
    var sum = 0;
    for (var i = 1; i <= Math.sqrt(num); i++) {
        var t1 = i * (num / i - i + 1); // adding i every time it appears with numbers greater than or equal to itself
        var t2 = (((num / i) * (num / i + 1)) / 2) - ((i * (i + 1)) / 2); // adding numbers that appear with i and are greater than i
        sum += t1 + t2;
    }
    return sum;
}
 
    var n;
    var sum = sum_all_divisors(n);
    document.write( sum );
     
// This code is contributed by shivanisinghss2110
</script>

Output

Time complexity: O(sqrt(N))

Auxiliary space: O(1)

Example in c:

Approach:

1. Create an array divisor_sum of size n+1, initialized with 1 for each index.
2. Loop through each prime p (starting from 2) and check if divisor_sum[p] is equal to 1. If so, update divisor_sum for each multiple of p using the formula: divisor_sum[i] *= (1 – pow(p, exp+1)) / (1 – p), where exp is the highest power of p that divides i.
3. For each prime p, add divisor_sum[p] to the variable sum.
4. Return sum.

C++

#include <bits/stdc++.h>
using namespace std;
 
int sum_of_divisors(int n) {
    int sum = 1;
    int divisor_sum[n+1];
    for (int i = 1; i <= n; i++) {
        divisor_sum[i] = 1;
    }
    for (int p = 2; p <= n; p++) {
        if (divisor_sum[p] == 1) {
            for (int i = p; i <= n; i += p) {
                int exp = 0;
                int q = i;
                while (q % p == 0) {
                    exp++;
                    q /= p;
                }
                divisor_sum[i] *= (1 - pow(p, exp+1)) / (1 - p);
            }
        }
        sum += divisor_sum[p];
    }
    return sum;
}
 
int main() {
    int n = 4;
    int sum = sum_of_divisors(n);
      cout<<"The sum of all divisors from 1 to "<<n<<" is "<<sum<<endl;
    return 0;
}

C

#include <stdio.h>
#include <math.h>
 
int power(int p,int exp)
{
  int res=1;
  for(int i=0;i<exp;i++)
  {
    res=res*p;
  }
  return res;
}
int sum_of_divisors(int n) {
    int sum = 1;
    int divisor_sum[n+1];
    for (int i = 1; i <= n; i++) {
        divisor_sum[i] = 1;
    }
    for (int p = 2; p <= n; p++) {
        if (divisor_sum[p] == 1) {
            for (int i = p; i <= n; i += p) {
                int exp = 0;
                int q = i;
                while (q % p == 0) {
                    exp++;
                    q /= p;
                }
                divisor_sum[i] *= (1 - power(p, exp+1)) / (1 - p);
            }
        }
        sum += divisor_sum[p];
    }
    return sum;
}
 
int main() {
    int n = 4;
    int sum = sum_of_divisors(n);
    printf("The sum of all divisors from 1 to %d is %d\n", n, sum);
    return 0;
}

Java

/*package whatever //do not write package name here */
 
import java.io.*;
 
class GFG {
  static int sum_of_divisors(int n) {
    int sum = 1;
    int divisor_sum[] = new int[n+1];
    for (int i = 1; i <= n; i++) {
        divisor_sum[i] = 1;
    }
    for (int p = 2; p <= n; p++) {
        if (divisor_sum[p] == 1) {
            for (int i = p; i <= n; i += p) {
                int exp = 0;
                int q = i;
                while (q % p == 0) {
                    exp++;
                    q /= p;
                }
                  int temp = (int)Math.pow(p,exp+1);
                divisor_sum[i]*= (1 - temp) / (1 - p);
            }
        }
        sum += divisor_sum[p];
    }
    return sum;
  }
  public static void main (String[] args) {
      int n = 4;
      int sum = sum_of_divisors(n);
      System.out.println("The sum of all divisors from 1 to "+n+" is "+sum);
  }
}
//this code is contributed by shubhamrajput6156

Python3

import math
 
def sum_of_divisors(n):
    sum = 1
    divisor_sum = [1] * (n+1)
    for i in range(2, n+1):
        if divisor_sum[i] == 1:
            for j in range(i, n+1, i):
                exp = 0
                q = j
                while q % i == 0:
                    exp += 1
                    q //= i
                divisor_sum[j] *= (1 - math.pow(i,exp+1)) // (1 - i)
        sum += divisor_sum[i]
    return sum
 
n = 4
sum = sum_of_divisors(n)
print(f"The sum of all divisors from 1 to {n} is {sum}")

C#

using System;
 
class Program {
    static int sum_of_divisors(int n) {
        int sum = 1;
        int[] divisor_sum = new int[n+1];
        for (int i = 1; i <= n; i++) {
            divisor_sum[i] = 1;
        }
        for (int p = 2; p <= n; p++) {
            if (divisor_sum[p] == 1) {
                for (int i = p; i <= n; i += p) {
                    int exp = 0;
                    int q = i;
                    while (q % p == 0) {
                        exp++;
                        q /= p;
                    }
                    divisor_sum[i] *= (1 - (int)Math.Pow(p, exp+1)) / (1 - p);
                }
            }
            sum += divisor_sum[p];
        }
        return sum;
    }
 
    static void Main(string[] args) {
        int n = 4;
        int sum = sum_of_divisors(n);
        Console.WriteLine("The sum of all divisors from 1 to " + n + " is " + sum);
    }
}

Javascript

function sum_of_divisors(n) {
  let sum = 1;
  let divisor_sum = new Array(n + 1).fill(1);
   
  for (let p = 2; p <= n; p++) {
    if (divisor_sum[p] === 1) {
      for (let i = p; i <= n; i += p) {
        let exp = 0;
        let q = i;
         
        while (q % p === 0) {
          exp++;
          q /= p;
        }
         
        let temp = Math.pow(p, exp + 1);
        divisor_sum[i] *= (1 - temp) / (1 - p);
      }
    }
     
    sum += divisor_sum[p];
  }
   
  return sum;
}
 
let n = 4;
let sum = sum_of_divisors(n);
console.log(`The sum of all divisors from 1 to ${n} is ${sum}`);

Output

The sum of all divisors from 1 to 4 is 15

The time complexity of this approach is O(n log log n) .

The space complexity is O(n) for the divisor_sum array.

Approach:

1. We initialize n to the maximum number till which we want to find the sum of divisors. In this example, we have taken n as 10.
2. We initialize an array of size n+1 to store the sum of divisors for each number from 1 to n.
3. We use two nested loops to iterate through all the numbers from 1 to n. The outer loop runs from 1 to n, while the inner loop runs from i to n in steps of i.
4. For each number j that is a multiple of i, we add i to the sum of divisors for j. This is done by accessing the array element divisors[j] and incrementing it by i.
5. Finally, we iterate through the array of sum of divisors and add up all the values to get the total sum of divisors from 1 to n.

C++

#include <bits/stdc++.h>
using namespace std;
#define MAXN 1000000
 
int main()
{
    int n = 10; // for example
    int i, j;
    int sum = 0;
    int divisors[MAXN+1] = {0}; // array to store the sum of divisors
 
    for (i = 1; i <= n; i++) {
        for (j = i; j <= n; j += i) {
            divisors[j] += i;
        }
    }
 
    for (i = 1; i <= n; i++) {
        sum += divisors[i];
    }
    cout<<"The sum of all divisors from 1 to "<<n<<" is "<<sum<<endl;
    return 0;
}

C

#include <stdio.h>
 
#define MAXN 1000000
 
int main()
{
    int n = 10; // for example
    int i, j;
    int sum = 0;
    int divisors[MAXN+1] = {0}; // array to store the sum of divisors
 
    for (i = 1; i <= n; i++) {
        for (j = i; j <= n; j += i) {
            divisors[j] += i;
        }
    }
 
    for (i = 1; i <= n; i++) {
        sum += divisors[i];
    }
 
    printf("The sum of all divisors from 1 to %d is %d\n", n, sum);
 
    return 0;
}

Java

import java.util.*;
 
public class Main {
    public static void main(String[] args) {
        int n = 10; // for example
        int i, j;
        int sum = 0;
        int[] divisors = new int[1000001]; // array to store the sum of divisors
 
        for (i = 1; i <= n; i++) {
            for (j = i; j <= n; j += i) {
                divisors[j] += i;
            }
        }
 
        for (i = 1; i <= n; i++) {
            sum += divisors[i];
        }
        System.out.println("The sum of all divisors from 1 to " + n + " is " + sum);
    }
}
// This code is contributed by shivhack999

Python3

MAXN = 1000000
 
n = 10  # for example
sum = 0
divisors = [0] * (MAXN+1)  # list to store the sum of divisors
 
for i in range(1, n+1):
    for j in range(i, n+1, i):
        divisors[j] += i
 
for i in range(1, n+1):
    sum += divisors[i]
 
print(f"The sum of all divisors from 1 to {n} is {sum}")

C#

using System;
 
class Program
{
    const int MAXN = 1000000;
 
    static void Main(string[] args)
    {
        int n = 10; // for example
        int i, j;
        int sum = 0;
        int[] divisors = new int[MAXN+1]; // array to store the sum of divisors
 
        for (i = 1; i <= n; i++) {
            for (j = i; j <= n; j += i) {
                divisors[j] += i;
            }
        }
 
        for (i = 1; i <= n; i++) {
            sum += divisors[i];
        }
 
        Console.WriteLine("The sum of all divisors from 1 to {0} is {1}", n, sum);
    }
}

Javascript

function main() {
    const n = 10; // for example
    let i, j;
    let sum = 0;
    const divisors = new Array(1000001).fill(0); // array to store the sum of divisors
 
    for (i = 1; i <= n; i++) {
        for (j = i; j <= n; j += i) {
            divisors[j] += i; // add the divisor 'i' to the sum of divisors for 'j'
        }
    }
 
    for (i = 1; i <= n; i++) {
        sum += divisors[i]; // add the sum of divisors for 'i' to the overall sum
    }
 
    console.log("The sum of all divisors from 1 to " + n + " is " + sum);
}
 
main(); // calling the main function to start the computation
 
//This code is contributed by uomkar369

Output

The sum of all divisors from 1 to 10 is 87

Time Complexity: O(nloglogn), where n is the maximum number till which we want to find the sum of divisors.
Space Complexity: O(n), as we are using an array of size n to store the sum of divisors.

Last Updated : 14 Aug, 2023