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 $nle x$, ErdH{o}s proved that $$C(x) sim e^{-gamma} x/logloglog{x}, quadtext{as $xtoinfty$}.$$ We show that $C(x)$ has an asymptotic series expansion, in the sense of Poincare, in descending powers of $logloglog{x}$, namely $$frac{e^{-gamma} x}{logloglog{x}} left(1-frac{gamma}{logloglog{x}} + frac{gamma^2 + frac{1}{12}pi^2}{(logloglog{x})^2} - frac{gamma^3 +frac{1}{4} gamma pi^2 + frac{2}{3}zeta(3)}{(logloglog{x})^3} + dots right). $$