The asymptotic formula for \(F_1(x)\) (Q1866889)

Let \(F_1(x)\) be the number of integers \(n\) not exceeding \(x\) and such that there exists exactly one (up to isomorphism) group of order \(n\). In 1948, P. Erdős established \(F_1(x)\sim e^{-\gamma}x/ \ln\ln x\), where \(\gamma=0,577\dots\) is the Euler constant. In this paper the following asymptotic formula is proved: \[ F_1(x)=\left(1+O\left(\frac {\ln\ln\ln\ln x}{\ln\ln\ln x}\right)\right)\frac {e^{-\gamma}x} {\ln\ln\ln x}. \]