For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we give a detailed exposition of the superior k-highly composite numbers. We then consider the function <Equation ID="Equ33"> <EquationSource Format="TEX">$\begin{aligned} f_k(n)=\frac{\log d_k(n)\log \log n}{\log k\log n},\quad n\ge 3 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>log</mo> <msub> <mi>d</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>log</mo> <mo>log</mo> <mi>n</mi> </mrow> <mrow> <mo>log</mo> <mi>k</mi> <mo>log</mo> <mi>n</mi> </mrow> </mfrac> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>which has a maximum value <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda (k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> at a superior k-highly composite number. We develop an efficient algorithm to compute <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda (k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the positive integer <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N_{\max }(k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mo movablelimits="true">max</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> achieves the value <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda (k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The results for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$2\le k\le 100$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mn>100</mn> </mrow> </math></EquationSource> </InlineEquation> are tabled.

Superior highly composite numbers and the explicit upper bound of generalized divisor functions

期刊论文
发表日期：

2025年12月17日

查看全文

Lee-Peng Teo

摘要

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