# Double Arithmetic Derivative

**Number Theory**Level 3

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

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 \(p\)

\((ab)'=a'b+ab'\)

\(0'=1'=0\)

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

The double arithmetic derivative, denoted by \(n''\), is simply defined by \(n''=(n')'\).

Find the sum of all positive integers \(n<100\) such that \(n''=1\)