  We show that if $N\pm 1=M\varphi(N)$ with $N\neq 15, 255$ composite, then
$M<15.76515\log\log\log N$ and $M<16.03235\log\log\omega(N)$, together with
similar results for the unitary totient function, Dedekind function, and the
sum of unitary divisors.
