Note that $$f(1) << f(2)<< f(4) << \cdots << f(1024)$$ This means $$f(1024)\ge 2f(512) \ge 4f(256) \ge \cdots \ge 1024f(1) \ge 1024$$ Therefore $M$ is at least $1024$. Equality can hold; for every $n \in \{1,2,\cdots , 2013\}$, take $f(n)=2^{a_1+a_2+\cdots + a_t}$, where $n=p_1^{a_1}p_2^{a_2} \cdots p_t^{a_t}$ is the canonical factorisation of $n$.