---

Claim. W. D. Banks and F. Luca, Noncototients and nonaliquots, arXiv:math/0409231v1 (14 September 2004). Its Theorem 1 proves that $2p$ is a noncototient, that is, not of the form $n-\phi(n)$, for all primes $p$ outside a set of relative density zero. Hence the number of noncototients up to $x$ is at least $(1+o(1))x/(2\log x)$. This answers yes to Problem 418, and the count improves the bound of order $\log x$ that the families of Browkin and Schinzel and of Flammenkamp and Luca give. The preprint's closing remarks combine Theorem 1 with Flammenkamp and Luca's criterion to raise the constant: the number of noncototients up to $x$ is at least $c(1+o(1))x/\log x$ for some $c>1/2$.

Depends on. No page of this wiki.

Standing. The journal version [BaLu05], Nonaliquots and Robbins numbers, Colloq. Math. 103 (2005), 27–32 (library card), keeps only the nonaliquot and Robbins-number theorems, so this result has no refereed or reviewed evidence. The site's discussion thread links the preprint (comment of 21 November 2025), but the site's commentary does not credit it.