# Minimum numbers needed to express every integer below N as a sum

We have an integer N. We need to express N as a sum of K integers such that by adding some(or all) of these integers we can get all the numbers in the range[1, N]. What is the minimum value of K?

**Examples:**

Input : N = 7 Output : 3 Explanation : Three integers are 1, 2, 4. By adding some(or all) of these groups we can get all number in the range 1 to N. 1; 2; 1+2=3; 4; 1+4=5; 2+4=6; 1+2+4=7 Input : N = 32 Output : 6 Explanation : Six integers are 1, 2, 4, 8, 16, 1.

1st we solve the problem for small numbers by hand.

n=1 : 1

n=2 : 1, 1

n=3 : 1, 2

n=4 : 1, 2, 1

n=5 : 1, 2, 2

n=6 : 1, 2, 3

n=7 : 1, 2, 4

n=8 : 1, 2, 4, 1

If we inspect this closely we can see that if then the integers are . Which is just another way of saying .So now we know for minimum value of K is m.

Now we inspect what happens for .For we just add a new integer 1 to our list of integers. Realize that for every number from we can increase the newly added integer by 1 and that will be the optimal list of integers. To verify look at N=4 to N=7, minimum K does not change; only the last integer is increased in each step.

Of course we can implement this in iterative manner in **O(log N)** time (by inserting successive powers of 2 in the list and the last element will be of the form N-(2^n-1)). But this is exactly same as finding the length of binary expression of N which also can be done in **O(log N)** time.

## C++

`// CPP program to find count of integers needed ` `// to express all numbers from 1 to N. ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// function to count length of binary expression of n ` `int` `countBits(` `int` `n) ` `{ ` ` ` `int` `count = 0; ` ` ` `while` `(n) { ` ` ` `count++; ` ` ` `n >>= 1; ` ` ` `} ` ` ` `return` `count; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 32; ` ` ` `cout << ` `"Minimum value of K is = "` ` ` `<< countBits(n) << endl; ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find count of integers needed ` `// to express all numbers from 1 to N ` ` ` `import` `java.io.*; ` ` ` `class` `GFG { ` ` ` `// function to count length of binary expression of n ` `static` `int` `countBits(` `int` `n) ` `{ ` ` ` `int` `count = ` `0` `; ` ` ` `while` `(n>` `0` `) { ` ` ` `count++; ` ` ` `n >>= ` `1` `; ` ` ` `} ` ` ` `return` `count; ` `} ` ` ` `// Driver code ` ` ` `public` `static` `void` `main (String[] args) { ` ` ` `int` `n = ` `32` `; ` ` ` `System.out.println(` `"Minimum value of K is = "` `+ ` ` ` `countBits(n)); ` ` ` ` ` `} ` `} ` |

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## Python 3

# Python3 program to find count of integers

# needed to express all numbers from 1 to N.

# function to count length of

# binary expression of n

def countBits(n):

count = 0;

while (n):

count += 1;

n >>= 1;

return count;

# Driver code

n = 32;

print(“Minimum value of K is =”,

countBits(n));

# This code is contributed by mits

## C#

`// C# program to find count of ` `// integers needed to express all ` `// numbers from 1 to N ` `using` `System; ` ` ` `class` `GFG ` `{ ` `// function to count length of ` `// binary expression of n ` `static` `int` `countBits(` `int` `n) ` `{ ` ` ` `int` `count = 0; ` ` ` `while` `(n > 0) ` ` ` `{ ` ` ` `count++; ` ` ` `n >>= 1; ` ` ` `} ` ` ` `return` `count; ` `} ` ` ` `// Driver code ` `static` `public` `void` `Main () ` `{ ` ` ` `int` `n = 32; ` ` ` `Console.WriteLine(` `"Minimum value of K is = "` `+ ` ` ` `countBits(n)); ` `} ` `} ` ` ` `// This code is contributed ` `// by Sach_Code ` |

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## PHP

`<?php ` `// PHP program to find count of integers ` `// needed to express all numbers from 1 to N. ` ` ` `// function to count length of ` `// binary expression of n ` `function` `countBits(` `$n` `) ` `{ ` ` ` `$count` `= 0; ` ` ` `while` `(` `$n` `) ` ` ` `{ ` ` ` `$count` `++; ` ` ` `$n` `>>= 1; ` ` ` `} ` ` ` `return` `$count` `; ` `} ` ` ` `// Driver code ` `$n` `= 32; ` `echo` `"Minimum value of K is = "` `, ` ` ` `countBits(` `$n` `), ` `"\n"` `; ` ` ` `// This code is contributed by Sachin ` `?> ` |

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**output:**

Minimum value of K is = 6

Please see count set bits for more efficient methods to count set bits in an integer.

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