Open. The site labels the problem OPEN and credits no solution; its remarks credit Berend and Osgood, and Bui, Pratt and Zaharescu, with density and counting bounds for the $n$ with $f(n)=m$, and Luca, under the abc conjecture, with $f(n)\to\infty$. The standing derives from the claim pages: the accepted partial claims Berend and Osgood 1992 and Bui, Pratt and Zaharescu 2023 are refereed bounds on how often $f(n)$ takes a given value; the accepted conditional claim Luca 2002 and the pending conditional claim Turturean 2026 rest on the unproved abc conjecture and derive nothing; so the problem is open with no settling or pending full claim.