One-Step $G$-Unimprovable Numbers

Abstract

The infinitude is established of the set ${\mathbf{U}}_1$ of positive integers $N>5$ such that $G(N)\le \min (G(N/q),G(Np))$ where $q,p$ are primes, $q|N$ 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 ${\mathbf{U}}_1$, the least of them $N_1^{\ast }=2^5\cdot 3^3\cdot 5^2\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23=160626866400,G(N_1^{\ast })=1.7374\dots$ Some interesting properties of these numbers are studied which may occur useful for the proof of Ramanujan-Robin inequality.

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