# Logarithmic Derivatives of Numbers

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...