Counting Distinct Solutions of a Diophantine Equation

Consider the Diophantine equation:

$$\frac{1}{x} + \frac{1}{y} = \frac{1}{n}, \quad (x,y,n \in \mathbb{N}^+)$$

For a given positive integer \(n\), your task is to count the number of essentially different solutions \( (x, y) \) (with \(x \le y\)) that satisfy the equation. For example, when \(n = 4\), there are three distinct solutions:

• $$\frac{1}{5} + \frac{1}{20} = \frac{1}{4}$$
• $$\frac{1}{6} + \frac{1}{12} = \frac{1}{4}$$
• $$\frac{1}{8} + \frac{1}{8} = \frac{1}{4}$$

By substituting \(x = n + a\) and \(y = n + b\), the equation reduces to \(a \times b = n^2\) (with \(a, b \in \mathbb{N}^+\)). Thus, the number of distinct solutions \( (x, y) \) with \(x \le y\) is equal to the number of pairs \((a, b)\) with \(a \times b = n^2\) and \(a \le b\). It can be shown that the answer is given by: $$\text{Answer} = \frac{d(n^2) + 1}{2},$$ where \(d(n^2)\) is the number of positive divisors of \(n^2\).

inputFormat

The input contains a single positive integer \(n\) on a single line.

outputFormat

Output a single integer representing the number of essentially different solutions \((x, y)\) (with \(x \le y\)) that satisfy the equation.

sample

4

3