jmc

Since $f(14)=7$, we know that $f(7)$ is defined, and it must equal $22$. Similarly, we know that $f(22)$ is defined, and it must equal $11$. Continuing on this way, $$\begin{array}{rl}f(11)& =34\\ f(34)& =17\\ f(17)& =52\\ f(52)& =26\\ f(26)& =13\\ f(13)& =40\\ f(40)& =20\\ f(20)& =10\\ f(10)& =5\\ f(5)& =16\\ f(16)& =8\\ f(8)& =4\\ f(4)& =2\\ f(2)& =1\\ f(1)& =4\end{array}$$We are now in a cycle $1$, $4$, $2$, $1$, and so on. Thus there are no more values which need to be defined, as there is no $a$ currently defined for which $f(a)$ is a $b$ not already defined. Thus the minimum number of integers we can define is the number we have already defined, which is $\boxed{18}$.