Double Arithmetic Derivative

ddxn0?\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 nn' denotes the arithmetic derivative of nn:

p=1p' = 1 for all primes pp

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

0=1=00'=1'=0

For example, 6=(2×3)=(2)(3)+(2)(3)=(1)(3)+(2)(1)=56'=(2\times3)'=(2')(3)+(2)(3')=(1)(3)+(2)(1)=5

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

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

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