Count of ways to represent N as sum of a prime number and twice of a square

Given an integer N, the task is to count the number of ways so that N can be written as the sum of a prime number and twice of a square, i.e.

, where P can be any prime number and A is any positive integer.
Note:

Examples:

Input: N = 9
Output:
Explanation:
9 can be represented as sum of prime number and twice a square in only one way –

Input: N = 15
Output:
Explanation:
15 can be represented as sum of prime number and twice a square in two ways –

[Tex]N = 15 = 13 + 2 * (1^{2})  [/Tex]

Approach: The idea is to use Seive of Eratosthenes to find all the primes and then for each prime number check for every possible number starting from 1. If any prime number and twice a square is equal to the given number then increment the count of the number of ways by 1.
Below is the implementation of the above approach:

C++

 `// C++ implementation to count the` `// number of ways a number can be` `// written as sum of prime number` `// and twice a square`   `#include `   `using` `namespace` `std;` `long` `long` `int` `n = 500000 - 2;` `vector<``long` `long` `int``> v;`   `// Function to mark all the` `// prime numbers using sieve` `void` `sieveoferanthones()` `{` `    ``bool` `prime[n + 1];`   `    ``// Intially all the numbers` `    ``// are marked as prime` `    ``memset``(prime, ``true``,` `           ``sizeof``(prime));`   `    ``// Loop to mark the prime numbers` `    ``// upto the Square root of N` `    ``for` `(``long` `long` `int` `i = 2; i <= ``sqrt``(n);` `         ``i++) {` `        ``if` `(prime[i])` `            ``for` `(``long` `long` `int` `j = i * i;` `                 ``j <= n; j += i) {` `                ``prime[j] = ``false``;` `            ``}` `    ``}`   `    ``// Loop to store the prime` `    ``// numbers in an array` `    ``for` `(``long` `long` `int` `i = 2; i < n; i++) {` `        ``if` `(prime[i])` `            ``v.push_back(i);` `    ``}` `}`   `// Function to find the number` `// ways to represent a number` `// as the sum of prime number and` `// square of a number` `void` `numberOfWays(``long` `long` `int` `n)` `{` `    ``long` `long` `int` `count = 0;`   `    ``// Loop to iterate over all the` `    ``// possible prime numbers` `    ``for` `(``long` `long` `int` `j = 1;` `         ``2 * (``pow``(j, 2)) < n; j++) {` `        ``for` `(``long` `long` `int` `i = 1;` `             ``v[i] + 2 <= n; i++) {`   `            ``// Incrment the count if` `            ``// the given number is a` `            ``// valid number` `            ``if` `(n == v[i] ` `+ (2 * (``pow``(j, 2))))` `                ``count++;` `        ``}` `    ``}` `    ``cout << count << endl;` `}`   `// Driver Code` `int` `main()` `{` `    ``sieveoferanthones();` `    ``long` `long` `int` `n = 9;`   `    ``// Function Call` `    ``numberOfWays(n);` `    ``return` `0;` `}`

Java

 `// Java implementation to count the` `// number of ways a number can be` `// written as sum of prime number` `// and twice a square` `import` `java.util.*;` `class` `GFG{`   `static` `int` `n = ``500000` `- ``2``;` `static` `Vector v = ` `              ``new` `Vector<>();`   `// Function to mark all the` `// prime numbers using sieve` `static` `void` `sieveoferanthones()` `{` `  ``boolean` `[]prime = ``new` `boolean``[n + ``1``];`   `  ``// Intially all the numbers` `  ``// are marked as prime` `  ``Arrays.fill(prime, ``true``);`   `  ``// Loop to mark the prime numbers` `  ``// upto the Square root of N` `  ``for` `(``int` `i = ``2``; ` `           ``i <= Math.sqrt(n); i++) ` `  ``{` `    ``if` `(prime[i])` `      ``for` `(``int` `j = i * i; ` `               ``j <= n; j += i) ` `      ``{` `        ``prime[j] = ``false``;` `      ``}` `  ``}`   `  ``// Loop to store the prime` `  ``// numbers in an array` `  ``for` `(``int` `i = ``2``; i < n; i++) ` `  ``{` `    ``if` `(prime[i])` `      ``v.add(i);` `  ``}` `}`   `// Function to find the number` `// ways to represent a number` `// as the sum of prime number and` `// square of a number` `static` `void` `numberOfWays(``int` `n)` `{` `  ``int` `count = ``0``;`   `  ``// Loop to iterate over all the` `  ``// possible prime numbers` `  ``for` `(``int` `j = ``1``; ``2` `* ` `      ``(Math.pow(j, ``2``)) < n; j++) ` `  ``{` `    ``for` `(``int` `i = ``1``; v.get(i) + ` `             ``2` `<= n; i++) ` `    ``{` `      ``// Incrment the count if` `      ``// the given number is a` `      ``// valid number` `      ``if` `(n == v.get(i) + ` `         ``(``2` `* (Math.pow(j, ``2``))))` `        ``count++;` `    ``}` `  ``}` `  ``System.out.print(count + ``"\n"``);` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `  ``sieveoferanthones();` `  ``int` `n = ``9``;`   `  ``// Function Call` `  ``numberOfWays(n);` `}` `}`   `// This code is contributed by Princi Singh`

Python3

 `# Python3 implementation to count the` `# number of ways a number can be` `# written as sum of prime number` `# and twice a square` `import` `math`   `n ``=` `500000` `-` `2` `v ``=` `[]`   `# Function to mark all the` `# prime numbers using sieve` `def` `sieveoferanthones():` `    `  `    ``prime ``=` `[``1``] ``*` `(n ``+` `1``)`   `    ``# Loop to mark the prime numbers` `    ``# upto the Square root of N` `    ``for` `i ``in` `range``(``2``, ``int``(math.sqrt(n)) ``+` `1``):` `        ``if` `(prime[i] !``=` `0``):` `            `  `            ``for` `j ``in` `range``(i ``*` `i, n ``+` `1``, i):` `                ``prime[j] ``=` `False` `            `  `    ``# Loop to store the prime` `    ``# numbers in an array` `    ``for` `i ``in` `range``(``2``, n):` `        ``if` `(prime[i] !``=` `0``):` `            ``v.append(i)` `    `  `# Function to find the number` `# ways to represent a number` `# as the sum of prime number and` `# square of a number` `def` `numberOfWays(n):` `    `  `    ``count ``=` `0`   `    ``# Loop to iterate over all the` `    ``# possible prime numbers` `    ``j ``=` `1` `    ``while` `(``2` `*` `(``pow``(j, ``2``)) < n):` `        ``i ``=` `1` `        ``while` `(v[i] ``+` `2` `<``=` `n):`   `            ``# Incrment the count if` `            ``# the given number is a` `            ``# valid number` `            ``if` `(n ``=``=` `v[i] ``+` `               ``(``2` `*` `(math.``pow``(j, ``2``)))):` `                ``count ``+``=` `1` `                `  `            ``i ``+``=` `1` `            `  `        ``j ``+``=` `1` `        `  `    ``print``(count)`   `# Driver Code` `sieveoferanthones()` `n ``=` `9`   `# Function call` `numberOfWays(n)`   `# This code is contributed by sanjoy_62`

C#

 `// C# implementation to count the` `// number of ways a number can be` `// written as sum of prime number` `// and twice a square         ` `using` `System; ` `using` `System.Collections;` `using` `System.Collections.Generic; `   `class` `GFG{         ` `            `  `static` `int` `n = 500000 - 2;`   `static` `ArrayList v = ``new` `ArrayList();`   `// Function to mark all the` `// prime numbers using sieve` `static` `void` `sieveoferanthones()` `{` `    ``bool` `[]prime = ``new` `bool``[n + 1];`   `    ``// Intially all the numbers` `    ``// are marked as prime` `    ``Array.Fill(prime, ``true``);`   `    ``// Loop to mark the prime numbers` `    ``// upto the Square root of N` `    ``for``(``int` `i = 2; ` `            ``i <= (``int``)Math.Sqrt(n); i++)` `    ``{` `        ``if` `(prime[i])` `        ``{` `            ``for``(``int` `j = i * i; ` `                    ``j <= n; j += i)` `            ``{` `                ``prime[j] = ``false``;` `            ``}` `        ``}` `    ``}`   `    ``// Loop to store the prime` `    ``// numbers in an array` `    ``for``(``int` `i = 2; i < n; i++)` `    ``{` `        ``if` `(prime[i])` `            ``v.Add(i);` `    ``}` `}`   `// Function to find the number` `// ways to represent a number` `// as the sum of prime number and` `// square of a number` `static` `void` `numberOfWays(``int` `n)` `{` `    ``int` `count = 0;`   `    ``// Loop to iterate over all the` `    ``// possible prime numbers` `    ``for``(``int` `j = 1; ` `            ``2 * (Math.Pow(j, 2)) < n; j++)` `    ``{` `        ``for``(``int` `i = 1; ` `           ``(``int``)v[i] + 2 <= n; i++) ` `        ``{` `            `  `            ``// Incrment the count if` `            ``// the given number is a` `            ``// valid number` `            ``if` `(n == (``int``)v[i] +` `                     ``(2 * (Math.Pow(j, 2))))` `                ``count++;` `        ``}` `    ``}` `    ``Console.Write(count);` `}         ` `        `  `// Driver Code         ` `public` `static` `void` `Main (``string``[] args)` `{         ` `    ``sieveoferanthones();` `    ``int` `n = 9;`   `    ``// Function call` `    ``numberOfWays(n);` `}         ` `}`   `// This code is contributed by rutvik_56`

Output:

```1

```

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