Logarithmic Arithmetic Derivative

Number Theory Level 5

\[\Large \frac{\frac{d}{dx}n}{n} \neq 0?\]

In calculus, when you take the derivative of a constant you get zero as an answer. In number theory, there is something called the arithmetic derivative which allows you to differentiate a number and get a nonzero answer. The arithmetic derivative works as follows.

Where \(n'\) denotes the arithmetic derivative of \(n\):

\(p' = 1\) for all primes \(p\)



For example, \(6'=(2\times3)'=(2')(3)+(2)(3')=(1)(3)+(2)(1)=5\)

Let us define the Logarithmic Arithmetic Derivative, denoted by \(L(n)\), as \(\frac{n'}{n}\). For how many ordered pairs \((a,b)\), where \(a\) and \(b\) are distinct positive integers less than \(100\), does \(L(a)=L(b)\)?

This is a member of a set of problems on the Arithmetic Derivative.

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