Superior Highly Composite Numbers and the Explicit Upper Bound of Generalized Divisor Functions

TL;DR

This paper explores superior k-highly composite numbers, introduces a function related to divisor counts, and develops an efficient algorithm to compute its maximum and the corresponding integers for k up to 100.

Contribution

It provides a detailed exposition of superior k-highly composite numbers and introduces an efficient algorithm to compute the maximum of a related divisor function.

Findings

Developed an efficient algorithm for computing λ(k) and N_max(k).

Tabulated results for 2 ≤ k ≤ 100.

Enhanced understanding of the structure of superior k-highly composite numbers.

Abstract

For $k\geq 2$, we give a detailed exposition of the superior $k$-highly composite numbers. We then consider the function \[f_k(n)=\frac{\log d_k(n)\log\log n}{\log k\log n},\quad n\geq 3\] which has a maximum value $\lambda(k)$ at a superior $k$-highly composite number. We develop an efficient algorithm to compute $\lambda(k)$ and the positive integer $N_{\max}(k)$ where $f_k$ achieves the value $\lambda(k)$. The results for $2\leq k\leq 100$ are tabled.