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Count total set bits in all numbers from 1 to N | Set 3
• Difficulty Level : Medium
• Last Updated : 11 Jun, 2021

Given a positive integer N, the task is to count the total number of set bits in binary representation of all the numbers from 1 to N.

Examples:

Input: N = 3
Output:
setBits(1) + setBits(2) + setBits(3) = 1 + 1 + 2 = 4

Input: N = 6
Output:

Approach: Solution to this problem has been published in the Set 1 and the Set 2 of this article. Here, a dynamic programming based approach is discussed.

• Base case: Number of set bits in 0 is 0.
• For any number n: n and n>>1 has same no of set bits except for the rightmost bit.

Example: n = 11 (1011),  n >> 1 = 5 (101)… same bits in 11 and 5 are marked bold. So assuming we already know set bit count of 5, we only need to take care for the rightmost bit of 11 which is 1. setBit(11) = setBit(5) + 1 = 2 + 1 = 3

Rightmost bit is 1 for odd and 0 for even number.

Recurrence Relation: setBit(n) = setBit(n>>1) + (n & 1) and setBit(0) = 0

We can use bottom-up dynamic programming approach to solve this.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;` `// Function to return the count of``// set bits in all the integers``// from the range [1, n]``int` `countSetBits(``int` `n)``{` `    ``// To store the required count``    ``// of the set bits``    ``int` `cnt = 0;` `    ``// To store the count of set``    ``// bits in every integer``    ``vector<``int``> setBits(n + 1);` `    ``// 0 has no set bit``    ``setBits = 0;` `    ``for` `(``int` `i = 1; i <= n; i++) {` `        ``setBits[i] = setBits[i >> 1] + (i & 1);``    ``}` `    ``// Sum all the set bits``    ``for` `(``int` `i = 0; i <= n; i++) {``        ``cnt = cnt + setBits[i];``    ``}` `    ``return` `cnt;``}` `// Driver code``int` `main()``{``    ``int` `n = 6;` `    ``cout << countSetBits(n);` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;` `class` `GFG``{` `// Function to return the count of``// set bits in all the integers``// from the range [1, n]``static` `int` `countSetBits(``int` `n)``{` `    ``// To store the required count``    ``// of the set bits``    ``int` `cnt = ``0``;` `    ``// To store the count of set``    ``// bits in every integer``    ``int` `[]setBits = ``new` `int``[n + ``1``];` `    ``// 0 has no set bit``    ``setBits[``0``] = ``0``;` `    ``// For the rest of the elements``    ``for` `(``int` `i = ``1``; i <= n; i++) {` `        ``setBits[i] = setBits[i >> ``1``] + (i & ``1``);``    ``}` `    ``// Sum all the set bits``    ``for` `(``int` `i = ``0``; i <= n; i++)``    ``{``        ``cnt = cnt + setBits[i];``    ``}``    ``return` `cnt;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `n = ``6``;` `    ``System.out.println(countSetBits(n));``}``}` `// This code is contributed by Princi Singh`

## Python3

 `# Python3 implementation of the approach` `# Function to return the count of``# set bits in all the integers``# from the range [1, n]``def` `countSetBits(n):` `    ``# To store the required count``    ``# of the set bits``    ``cnt ``=` `0` `    ``# To store the count of set``    ``# bits in every integer``    ``setBits ``=` `[``0` `for` `x ``in` `range``(n ``+` `1``)]` `    ``# 0 has no set bit``    ``setBits[``0``] ``=` `0` `    ``# For the rest of the elements``    ``for` `i ``in` `range``(``1``, n ``+` `1``):``        ``setBits[i] ``=` `setBits[i ``/``/` `2``] ``+` `(i & ``1``)` `    ``# Sum all the set bits``    ``for` `i ``in` `range``(``0``, n ``+` `1``):``        ``cnt ``=` `cnt ``+` `setBits[i]``    ` `    ``return` `cnt` `# Driver code``n ``=` `6``print``(countSetBits(n))` `# This code is contributed by Sanjit Prasad`

## C#

 `// C# implementation of the approach``using` `System;` `class` `GFG``{``    ` `    ``// Function to return the count of``    ``// set bits in all the integers``    ``// from the range [1, n]``    ``static` `int` `countSetBits(``int` `n)``    ``{``    ` `        ``// To store the required count``        ``// of the set bits``        ``int` `cnt = 0;``    ` `        ``// To store the count of set``        ``// bits in every integer``        ``int` `[]setBits = ``new` `int``[n + 1];``    ` `        ``// 0 has no set bit``        ``setBits = 0;``    ` `        ``// 1 has a single set bit``        ``setBits = 1;``    ` `        ``// For the rest of the elements``        ``for` `(``int` `i = 2; i <= n; i++)``        ``{``    ` `            ``// If current element i is even then``            ``// it has set bits equal to the count``            ``// of the set bits in i / 2``            ``if` `(i % 2 == 0)``            ``{``                ``setBits[i] = setBits[i / 2];``            ``}``    ` `            ``// Else it has set bits equal to one``            ``// more than the previous element``            ``else``            ``{``                ``setBits[i] = setBits[i - 1] + 1;``            ``}``        ``}``    ` `        ``// Sum all the set bits``        ``for` `(``int` `i = 0; i <= n; i++)``        ``{``            ``cnt = cnt + setBits[i];``        ``}``        ``return` `cnt;``    ``}``    ` `    ``// Driver code``    ``static` `public` `void` `Main ()``    ``{``        ``int` `n = 6;``    ` `        ``Console.WriteLine(countSetBits(n));``    ``}``}` `// This code is contributed by AnkitRai01`

## Javascript

 ``
Output:
`9`

Another simple and easy to understand solution:

A simple easy to implement and understand solution would be not using bits operations.  The solution is to directly count set bits using __builtin_popcount() .The solution is explained in code using comments.

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;` `// Function to return the count of``// set bits in all the integers``// from the range [1, n]``int` `countSetBits(``int` `n)``{` `    ``// To store the required count``    ``// of the set bits``    ``int` `cnt = 0;` `    ``// Calculate set bits in each number using``    ``// __builtin_popcount() and  Sum all the set bits``    ``for` `(``int` `i = 1; i <= n; i++) {``        ``cnt = cnt + __builtin_popcount(i);``    ``}` `    ``return` `cnt;``}` `// Driver code``int` `main()``{``    ``int` `n = 6;` `    ``cout << countSetBits(n);` `    ``return` `0;``}` `// This article is contributed by Abhishek`
Output
`9`

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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