# Find all numbers up to N which are both Pentagonal and Hexagonal

Given an integer N, the task is to find all numbers up to N, which are both Pentagonal as well as Hexagonal.

Example:

Input: N = 1000
Output: 1

Input: N = 100000
Output: 1, 40755

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

• To solve the problem, we are generating all pentagonal numbers up to N and check if those are hexagonal numbers or not.
• Formula to calculate ith Pentagonal Number:

i * ( 3 * i – 1 ) / 2

• To check if a pentagonal number, say pn, is a hexagonal number or not:

( 1 + sqrt(8 * pn + 1 ) ) / 4 needs to be a Natural number

Below is the implementation of the above approach:

## C++

 `// C++ Program of the above approach  ` `#include   ` `using` `namespace` `std;  ` ` `  `// Function to print numbers upto N  ` `// which are both pentagonal as well  ` `// as hexagonal numbers  ` `void` `pen_hex(``long` `long` `n)  ` `{  ` `    ``long` `long` `pn = 1;  ` `    ``for` `(``long` `long` `int` `i = 1;; i++) {  ` ` `  `        ``// Calculate i-th pentagonal number  ` `        ``pn = i * (3 * i - 1) / 2;  ` ` `  `        ``if` `(pn > n)  ` `            ``break``;  ` ` `  `        ``// Check if the pentagonal number  ` `        ``// pn is hexagonal or not  ` `        ``long` `double` `seqNum  ` `            ``= (1 + ``sqrt``(8 * pn + 1)) / 4;  ` ` `  `        ``if` `(seqNum == ``long``(seqNum))  ` `            ``cout << pn << ``", "``;  ` `    ``}  ` `}  ` ` `  `// Driver Program  ` `int` `main()  ` `{  ` `    ``long` `long` `int` `N = 1000000;  ` `    ``pen_hex(N);  ` `    ``return` `0;  ` `}  `

## Java

 `// Java program of the above approach  ` `import` `java.util.*;  ` ` `  `class` `GFG{  ` ` `  `// Function to print numbers upto N  ` `// which are both pentagonal as well  ` `// as hexagonal numbers  ` `static` `void` `pen_hex(``long` `n)  ` `{  ` `    ``long` `pn = ``1``;  ` `    ``for``(``long` `i = ``1``; i < n; i++)  ` `    ``{  ` `         `  `        ``// Calculate i-th pentagonal number  ` `        ``pn = i * (``3` `* i - ``1``) / ``2``;  ` ` `  `        ``if` `(pn > n)  ` `            ``break``;  ` ` `  `        ``// Check if the pentagonal number  ` `        ``// pn is hexagonal or not  ` `        ``double` `seqNum = (``1` `+ Math.sqrt(  ` `                        ``8` `* pn + ``1``)) / ``4``;  ` ` `  `        ``if` `(seqNum == (``long``)seqNum)  ` `            ``System.out.print(pn + ``", "``);  ` `    ``}  ` `}  ` ` `  `// Driver code  ` `public` `static` `void` `main(String[] args)  ` `{  ` `    ``long` `N = ``1000000``;  ` `    ``pen_hex(N);  ` `}  ` `}  ` ` `  `// This code is contributed by offbeat  `

## Python3

 `# Python3 program of the above approach ` `import` `math ` ` `  `# Function to print numbers upto N  ` `# which are both pentagonal as well  ` `# as hexagonal numbers  ` `def` `pen_hex(n):  ` ` `  `    ``pn ``=` `1` `    ``for` `i ``in` `range``(``1``, N):  ` ` `  `        ``# Calculate i-th pentagonal number  ` `        ``pn ``=` `(``int``)(i ``*` `(``3` `*` `i ``-` `1``) ``/` `2``)  ` ` `  `        ``if` `(pn > n): ` `            ``break` ` `  `        ``# Check if the pentagonal number  ` `        ``# pn is hexagonal or not  ` `        ``seqNum ``=` `(``1` `+` `math.sqrt(``8` `*` `pn ``+` `1``)) ``/` `4` ` `  `        ``if` `(seqNum ``=``=` `(``int``)(seqNum)): ` `            ``print``(pn, end ``=` `", "``) ` ` `  `# Driver Code ` `N ``=` `1000000` ` `  `pen_hex(N) ` ` `  `# This code is contributed by divyeshrabadiya07 `

## C#

 `// C# program of the above approach   ` `using` `System;  ` `using` `System.Collections.Generic;  ` ` `  `class` `GFG{          ` `             `  `// Function to print numbers upto N ` `// which are both pentagonal as well ` `// as hexagonal numbers ` `static` `void` `pen_hex(``long` `n) ` `{ ` `    ``long` `pn = 1; ` `    ``for``(``long` `i = 1;; i++) ` `    ``{ ` `         `  `        ``// Calculate i-th pentagonal number ` `        ``pn = i * (3 * i - 1) / 2; ` ` `  `        ``if` `(pn > n) ` `            ``break``; ` ` `  `        ``// Check if the pentagonal number ` `        ``// pn is hexagonal or not ` `        ``double` `seqNum = (1 + Math.Sqrt( ` `                         ``8 * pn + 1)) / 4; ` ` `  `        ``if` `(seqNum == (``long``)(seqNum)) ` `        ``{ ` `            ``Console.Write(pn + ``", "``); ` `        ``} ` `    ``} ` `}  ` `         `  `// Driver Code          ` `public` `static` `void` `Main (``string``[] args) ` `{          ` `    ``long` `N = 1000000; ` `     `  `    ``pen_hex(N); ` `}          ` `} ` ` `  `// This code is contributed by rutvik_56 `

Output:

```1, 40755,
```

Time Complexity: O(N)
Auxiliary space: O(1)

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