For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le(2\log2+o(1))\,n/\log n$. This improves the bounds of order $n/(\log n)^{1/2-\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above $\sqrt{n!}$, with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant $2\log2$.
Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erd\H{o}s, Graham, Ivi\'c and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n)>(1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a facto...
For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding...
Denote by $\omega(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(\omega(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $\omega_{z}(n...
T. Minamide, Haruka Sakai, Y. Tanigawa· 1 citation
For an integer $n>1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for...
Following Erd\H{o}s (1982) and Sanna (2019), we study the arithmetic function $h(n)$, which is defined to be the number of distinct exponents in the prime factorization of a positive integer $n$. Among other things, we show that $$ \sum_{n\leq x}h(\phi(n)) \asymp x\left(\frac{\log\log x}{\log\log\log x}\right)^{1/2}, $...
Mikhail R. Gabdullin, V. V. Iudelevich· 0 citations
We prove that for every integer $N\geq2$, there exist positive integers $a$ and $b$ such that $N=a+b$ and $\Omega(ab)\leq33$, where $\Omega(n)$ denotes the number of prime factors of $n$, counted with multiplicity. This improves the previous bound of $40$ obtained by Dudek and Dunn. The proof applies the explicit Fried...
Peter J. Campbell· 0 citations
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