Sum of all second largest divisors after splitting a number into one or more parts

Given an integer **N**( 2 <= N <= 10^9 ), split the number into one or more parts(possibly none), where each part must be greater than 1. The task is to find the minimum possible sum of the second largest divisor of all the splitting numbers.

**Examples:**

Input :N = 27Output :3Explanation :Split the given number into 19, 5, 3. Second largest divisor of each number is 1. So, sum is 3.Input :N = 19Output :1Explanation :Don't make any splits. Second largest divisor of 19 is 1. So, sum is 1

**Approach: **

The idea is based on Goldbach’s conjecture.

- When the number is prime, then the answer will be 1.
- When a number is even then it can always be expressed as a sum of 2 primes. So, the answer will be 2.
- When the number is odd,
- When N-2 is prime, then the number can be express as the sum of 2 primes, that are 2 and N-2, then the answer will be 2.
- Otherwise, the answer will always be 3.

Below is the implementation of the above approach:

## C++

`// CPP program to find sum of all second largest divisor ` `// after splitting a number into one or more parts ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find a number is prime or not ` `bool` `prime(` `int` `n) ` `{ ` ` ` `if` `(n == 1) ` ` ` `return` `false` `; ` ` ` ` ` `// If there is any divisor ` ` ` `for` `(` `int` `i = 2; i * i <= n; ++i) ` ` ` `if` `(n % i == 0) ` ` ` `return` `false` `; ` ` ` ` ` `return` `true` `; ` `} ` ` ` `// Function to find the sum of all second largest divisor ` `// after splitting a number into one or more parts ` `int` `Min_Sum(` `int` `n) ` `{ ` ` ` `// If number is prime ` ` ` `if` `(prime(n)) ` ` ` `return` `1; ` ` ` ` ` `// If n is even ` ` ` `if` `(n % 2 == 0) ` ` ` `return` `2; ` ` ` ` ` `// If the number is odd ` ` ` `else` `{ ` ` ` ` ` `// If N-2 is prime ` ` ` `if` `(prime(n - 2)) ` ` ` `return` `2; ` ` ` ` ` `// There exists 3 primes x1, x2, x3 ` ` ` `// such that x1 + x2 + x3 = n ` ` ` `else` ` ` `return` `3; ` ` ` `} ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 27; ` ` ` ` ` `// Function call ` ` ` `cout << Min_Sum(n); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to Sum of all second largest ` `// divisors after splitting a number into one or more parts ` `import` `java.io.*; ` ` ` `class` `GFG { ` ` ` ` ` ` ` `// Function to find a number is prime or not ` `static` `boolean` `prime(` `int` `n) ` `{ ` ` ` `if` `(n == ` `1` `) ` ` ` `return` `false` `; ` ` ` ` ` `// If there is any divisor ` ` ` `for` `(` `int` `i = ` `2` `; i * i <= n; ++i) ` ` ` `if` `(n % i == ` `0` `) ` ` ` `return` `false` `; ` ` ` ` ` `return` `true` `; ` `} ` ` ` `// Function to find the sum of all second largest divisor ` `// after splitting a number into one or more parts ` `static` `int` `Min_Sum(` `int` `n) ` `{ ` ` ` `// If number is prime ` ` ` `if` `(prime(n)) ` ` ` `return` `1` `; ` ` ` ` ` `// If n is even ` ` ` `if` `(n % ` `2` `== ` `0` `) ` ` ` `return` `2` `; ` ` ` ` ` `// If the number is odd ` ` ` `else` `{ ` ` ` ` ` `// If N-2 is prime ` ` ` `if` `(prime(n - ` `2` `)) ` ` ` `return` `2` `; ` ` ` ` ` `// There exists 3 primes x1, x2, x3 ` ` ` `// such that x1 + x2 + x3 = n ` ` ` `else` ` ` `return` `3` `; ` ` ` `} ` `} ` ` ` `// Driver code ` ` ` ` ` ` ` `public` `static` `void` `main (String[] args) { ` ` ` `int` `n = ` `27` `; ` ` ` ` ` `// Function call ` ` ` `System.out.println( Min_Sum(n)); ` ` ` `} ` `} ` ` ` `// This code is contributed by anuj_6 ` |

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## Python3

`# Python 3 program to find sum of all second largest divisor ` `# after splitting a number into one or more parts ` ` ` `from` `math ` `import` `sqrt ` `# Function to find a number is prime or not ` `def` `prime(n): ` ` ` `if` `(n ` `=` `=` `1` `): ` ` ` `return` `False` ` ` ` ` `# If there is any divisor ` ` ` `for` `i ` `in` `range` `(` `2` `,` `int` `(sqrt(n))` `+` `1` `,` `1` `): ` ` ` `if` `(n ` `%` `i ` `=` `=` `0` `): ` ` ` `return` `False` ` ` ` ` `return` `True` ` ` `# Function to find the sum of all second largest divisor ` `# after splitting a number into one or more parts ` `def` `Min_Sum(n): ` ` ` `# If number is prime ` ` ` `if` `(prime(n)): ` ` ` `return` `1` ` ` ` ` `# If n is even ` ` ` `if` `(n ` `%` `2` `=` `=` `0` `): ` ` ` `return` `2` ` ` ` ` `# If the number is odd ` ` ` `else` `: ` ` ` `# If N-2 is prime ` ` ` `if` `(prime(n ` `-` `2` `)): ` ` ` `return` `2` ` ` ` ` `# There exists 3 primes x1, x2, x3 ` ` ` `# such that x1 + x2 + x3 = n ` ` ` `else` `: ` ` ` `return` `3` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `'__main__'` `: ` ` ` `n ` `=` `27` ` ` ` ` `# Function call ` ` ` `print` `(Min_Sum(n)) ` ` ` `# This code is contributed by ` `# Surendra_Gangwar ` |

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## C#

`// C# program to Sum of all second largest ` `// divisors after splitting a number into one or more parts ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `// Function to find a number is prime or not ` `static` `bool` `prime(` `int` `n) ` `{ ` ` ` `if` `(n == 1) ` ` ` `return` `false` `; ` ` ` ` ` `// If there is any divisor ` ` ` `for` `(` `int` `i = 2; i * i <= n; ++i) ` ` ` `if` `(n % i == 0) ` ` ` `return` `false` `; ` ` ` ` ` `return` `true` `; ` `} ` ` ` `// Function to find the sum of all second largest divisor ` `// after splitting a number into one or more parts ` `static` `int` `Min_Sum(` `int` `n) ` `{ ` ` ` `// If number is prime ` ` ` `if` `(prime(n)) ` ` ` `return` `1; ` ` ` ` ` `// If n is even ` ` ` `if` `(n % 2 == 0) ` ` ` `return` `2; ` ` ` ` ` `// If the number is odd ` ` ` `else` `{ ` ` ` ` ` `// If N-2 is prime ` ` ` `if` `(prime(n - 2)) ` ` ` `return` `2; ` ` ` ` ` `// There exists 3 primes x1, x2, x3 ` ` ` `// such that x1 + x2 + x3 = n ` ` ` `else` ` ` `return` `3; ` ` ` `} ` `} ` ` ` `// Driver code ` `public` `static` `void` `Main () ` `{ ` ` ` `int` `n = 27; ` ` ` ` ` `// Function call ` ` ` `Console.WriteLine( Min_Sum(n)); ` `} ` `} ` ` ` `// This code is contributed by anuj_6 ` |

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**Output:**

3

**Time complexity:** O(sqrt(N))

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