Logarithmic Derivatives of Numbers

Number Theory Level 5

The number-theoretical derivative \(Dn\) of a natural number \(n\) is defined recursively by the rule \[Dp = 1\ \text{for prime}\ p;\ \ \ D(a\cdot b) = a\cdot Db + Da\cdot b.\] (Note the analogy to the product rule for derivatives in calculus.)

From this definition it follows, for instance, that if \(n = p^a\), then \(Dn = ap^{a-1}\). An other concrete example is \[D(18) = D(2\cdot 9) = 2\cdot D9 + D2\cdot 9 \\ = 2\cdot 6 + 1\cdot 9 = 12 + 9 = 21.\]

We take number-theoretical derivatives to the next level and define the logarithmic derivative \[Ln = \frac{Dn}n.\] (We call this "logarithmic derivative" based on the Calculus equality, \(d(\ln f(x))/dx = (df(x)/dx)/x\).)

For instance, \[L(18) = \frac{D(18)}{18} = \frac{21}{18} = \frac{7}{6}.\]

Question: What are the two smallest distinct natural numbers \(a, b\) for which \(La = Lb\)? Give your answer as the sum \(a + b\).


Problem Loading...

Note Loading...

Set Loading...