The values of the Euler phi function $\phi(n)$ are tabulated at OEIS A$000010$.
Each of these values is even except for $\phi(1) = \phi(2) = 1$ . However, not every even number arises in this way. Those that don't $( 14, 26, 34, 38, ... )$ are listed at OEIS A$005277$ .
A little experimentation suggests that every even number can be written as
$$2n = \phi(a) + \phi(b)$$
If we invoke Goldbach's conjecture, then this can be shown as follows. Writing $2n + 2 = p + q$ for primes $p$ and $q$ yields $2n = \phi(p) + \phi(q)$ . [For odd numbers we could rewrite this using three summands as $2n +1 = \phi(p) + \phi(q) + \phi(1)$ . ]
Question: There are many more phi values than (shifted) primes indicating that this problem might be accessible even though Goldbach currently is not. Is it possible to give an unconditional proof that $2n = \phi(a) + \phi(b)$ always has a solution?