Sreejato's ordered pairs

Number Theory Level 5

For positive integers \(x\), let \(f(x)\) be the number of ordered pairs of positive integers \( (a, b) \) such that \[ \frac{1}{a} + \frac{1}{b} = \frac{1}{x}. \]

Find the smallest possible value of \( N \) such that \(N\) is a prime power and \( f(N)= 13 \).

This problem is posed by Sreejato B.

Details and assumptions

A prime power is of the form \( p^n \), where \( p \) is a prime number and \( n \) is a positive integer.


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