Claims

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1994_04_11_browkin_schinzel: Proves that no number 2^k times 509203 with k at least 1 is of the form n minus Euler's totient of n, so infinitely many positive integers are not of that form; refereed in Colloquium Mathematicum and recorded by Guy.

1999_07_02_flammenkamp_luca: Gives a sufficient condition on k for every 2^m k with m at least 1 to be a noncototient and finds seven such k by computation, so infinitely many integers are not of the form n minus phi(n); refereed.

2004_09_14_banks_luca: Proves that 2p is a noncototient for almost all primes p, so at least (1+o(1))x/(2 log x) integers up to x are not of the form n minus phi(n); arXiv preprint only, the journal version omitting this theorem.

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