Erdős problem 897

Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$ such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$. Is it true that $\limsup_n (f(n+1)−f(n))/ \log n = ∞$?