Sum of all divisors from 1 to n

Difficulty Level : Medium
Last Updated : 05 Jan, 2021

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.*/

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 comtributed 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.*/

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;
    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

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)

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