Numbers which are orders only of cyclic groups

We call n a cyclic number if every group of order n is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that n is cyclic precisely when \gcd (n,\phi (n))=1. With C(x) denoting the count of cyclic n\le x, Erdős proved that \[ C(x) \sim e^{-\gamma } x/\!\log \log \log {x}, \quad \text {as x\to \infty }. \] We show that C(x) has an asymptotic series expansion, in the sense of Poincaré, in descending powers of \log \log \log {x}, namely \[ \frac {e^{-\gamma } x}{\log \log \log {x}} \left (1\!-\!\frac {\gamma }{\log \log \log {x}} \!+\! \frac {\gamma ^2 + \frac {1}{12}\pi ^2}{(\log \log \log {x})^2} \!-\! \frac {\gamma ^3 +\frac {1}{4} \gamma \pi ^2 \!+\! \frac {2}{3}\zeta (3)}{(\log \log \log {x})^3} + \dots \right ). \]