One-Step $G$-Unimprovable Numbers

The infinitude is established of the set ${\mathbf{U}}_{\mathbf{1}}$ of positive integers $N>5$ such that $G(N)\le \min (G(N\mathrm{/}q),G(Np))$ where $q,p$ are primes, $q\mathrm{\mid }N$ and $G(N):=\frac{\sigma (N)}{N\log \log N}$ stands for Gronwall number, $\sigma (N)$ being the sum of all divisors of $N$. The constructive algorithm is proposed which successively calculates the elements of ${\mathbf{U}}_{\mathbf{1}}$, the least of them $N_1^=2^5\cdot 3^3 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 =160\ 626\ 866\ 400,
G(N_1^)=1.7374\dots$ Some interesting properties of these numbers are studied which may occur useful for the proof of Ramanujan-Robin inequality.