Fractional parts of \(\log p\) and a digit function (Q1311340)

For \(n = p^{r_ 1}_ 1\dots p^{r_ K}_ K\) a positive integer, let \(s(n):=\sum^ K_{i=1} r_ i D(p_ i) - D(n)\) where \(D(m)\) denotes the number of digits of an integer \(m\) in its decimal expansion. For integers \(K\), \(r_ 1,\dots,r_ K \geq 1\), a positive integer \(n\) is called to be of type \((K;r_ 1,\dots,r_ K)\) if \(n = p^{r_ 1}_ 1\dots p^{r_ K}_ K\) for some distinct primes \(p_ 1,\dots,p_ K\). It is shown that for \(m\geq 0\) there is an \(n\) of type \((K;r_ 1,\dots,r_ K)\) in \(s^{-1}(m)\) if and only if \(\sum^ K_{i=1}r_ i\geq m+1\).