Are there infinitely many integers not of the form $n - \phi(n)$?

It follows from a slight strengthening of the Goldbach conjecture that every odd number can be written as $n - \phi(n)$. In particular, we assume that every even number greater than 6 can be written as the sum of two *distinct* primes, in contrast to the usual Goldbach conjecture that every even number greater than 2 can be written as the sum of two primes.