On locally repeated values of certain arithmetic functions. I (Q1063027)

It is shown that, for certain integer-valued arithmetic functions \(f\), the equation \(n+f(n)=m+f(m)\) has infinitely many solutions with \(n\neq m\). Let \(\nu(n)\) denote the number of distinct prime factors of \(n\). Then, for \(f=\nu\), a lower bound for the number of solutions \(n,m\leq x\) is given.