On a conjecture of Deaconescu

In 2000 Deaconescu raised a question whether there exists a composite $n$ for which $S_2(n)\mathrm{\mid }\phi (n)-1$, where $\phi (n)$ is Euler’s function and $S_2(n)$ is Schemmel’s totient function. In this paper we prove that any such $n$ is odd, squarefree and has at least seven distinct prime factors. We also prove that any such $n$ with exactly $K$ distinct prime divisors is necessarily less than $2^{2^{K+1}}$.