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{align*} 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{align*}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} $.