Is $Bigl{ n sum_{k=2}^{n-1} frac{1}{k}Bigr}$ unique $forall n in Bbb{N}, n>1$

If we define $$f(n) = Bigl{ n sum_{k=2}^{n-1} frac{1}{k}Bigr}$$

is it true that $f(n) ne f(m)$ whenever $n ne m, forall m,n in Bbb{N}$ (where the curly braces denote the fractional part)?

I wanted to explore coming up with a kind of "global residue" concept for a number, based on all the numbers less than it.