Miklos Bona, Bruce Sagan
Using Kummer's Theorem, we give a necessary and sufficient condition for a Narayana number to be divisible by a given prime. We use this to derive certain properties of the Narayana triangle.
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Divisibility of character values of the symmetric group by prime powers
本文证明了Miller猜想，表明当n趋于无穷时对称群S_n特征表的几乎所有元素都能被任意给定素数幂整除，推广了此前关于素数整除性的结果。
Wall published a paper in 1960 on the Fibonacci sequence where he derived many results concerning the period and prime power divisibility modulo m. His periodicity results have been generalized to second order linear recurrences. Here we study the sequences generated by such recurrences, with starting values of {0,1}: among other things, we derive new prime power divisibility results, derive the period by new methods, establish new identities, show derivations involving powers of matrices generated by these general recurrences, etc.
For an arbitrary prime $p$ we prove that the proportion of entries divisible by $p$ in certain columns of the character table of the symmetric group $S_n$ tends to 1 as $n\to\infty$. This is done by finding lower bounds on the number of $k$-cores, for $k$ large enough with respect to $n$.
We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is $1+\log n / \log p_j$; and our theorem shows that for every $\varepsilon > 0$, $r \geq 1$, and any sufficiently large primes $p_1,...,p_r > p_0(\varepsilon,r)$, we can find integers $n$ where for $j=1,...,r$, $p_j$ divides ${2n \choose n}$ with multipli…
Divisibility properties of weighted $k$ regular partitions
本文研究了加权 $k$ 正则划分的可除性性质，推导了新的同余式、可除性结果和正密度素数集。
We study the undirected divisibility graph in which the vertex set is a finite subset of consecutive natural numbers up to N.We derive analytical expressions for measures of the graph like degree, clustering, geodesic distance and centrality in terms of the floor functions and the divisor functions. We discuss how these measures depend on the vertex labels and the size of graph N. We also present the specific case of prime vertices separately as corollaries. We could explain the patterns in the local measures for a finite size graph as well as the trends in global measures as the size of the g…
Notes on the Divisibility of Divisor Sums of Catalan Numbers
本文研究卡塔兰数的约数和函数σ(C_n)的可除性，建立6k-1型素因子条件与可除性判据，并用de Bruijn定理给出渐近估计，揭示其算术结构。
In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let $n>2$ and $q>0$ be integers with $2\mid n$ or $2\nmid q$. We show that $$\sum_{k=0}^{p-1}\frac{(q-\frac{p}{n})_k^n}{(1)_k^n}\equiv0\pmod{p^3} $$ and $$p^n\sum_{k=0}^{p-1}\frac{(1)_k^n}{(\frac{p}{n}-q+2)_k^n}\equiv0\pmod{p^3}$$ for any prime $p>\max\{n,(q-1)n+1\}$, where $(x)_k$ denotes the Pochhammer symbol defined by $$ (x)_k=\begin{cases}1,\quad &k=0,\\ x(x+1)\cdots(x+k-1),\quad &k>0.\end{cases}$$ Let $n\geq4$ be an even integer. Then for a…
We examine an elementary problem on prime divisibility of binomial coefficients. Our problem is motivated by several related questions on alternating groups.
Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers
将Midy定理推广到非整数基的计数系统，研究分数在基β>1下的展开性质，并刻画了基为黄金比例时满足Midy性质的素数分母q。
We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base~$q$ and the $A_j$'s are positive integers relatively prime to $q$. We do this in fact within the framework of quasi-additive functions. As application, we show that most elements of "Catalan-like" sequences - by which we mean integer sequences defined by products/quotients of factorials - are divisible by any given positive integer.
In 1982, Gessel showed that the Apéry numbers associated to the irrationality of $ζ(3)$ satisfy Lucas congruences. Our main result is to prove corresponding congruences for all sporadic Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol--van Straten and Rowland--Yassawi to establish these congruences. However, for the sequences often labeled $s_{18}$ and $(η)$ we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist--Zudilin numbers are periodic modulo …
Average divisibility in character tables of $\mathrm{GL}_2(\mathbb{F}_q)$
研究 $\mathrm{GL}_2(\mathbb{F}_q)$ 特征标表中不被固定素数 $\ell$ 整除的条目比例，证明该比例趋于 $1/2$，而非零条目中不被 $\ell$ 整除的比例趋于 $1$，并证明非零特征值的角度在 $[0,2\pi]$ 上均匀分布。
We mainly introduce two new kinds of numbers given by $$R_n=\sum_{k=0}^n\binom nk\binom{n+k}k\frac1{2k-1}\quad\ (n=0,1,2,...)$$ and $$S_n=\sum_{k=0}^n\binom nk^2\binom{2k}k(2k+1)\quad\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\equiv1\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\equiv1\pmod 4$ and $y\equiv0\pmod 2$), then $$R_{(p-1)/2}\equiv p-(-1)^{(p-1)/4}2x\pmod{p^2}.$$ Also, $$\frac1{n^2}\sum_{k=0}^{n-1}S_k\in\mathbb Z\ \ {and}\ \ \frac1n\sum_{k=0}^{n-1}S_k(x)\in\mathbb Z[x]\quad\text{for all}\ n=1,2,3,...,$$ where $S_k(x)=\s…
The Divisibility of $\mathrm{GL}(n, q)$ Character Values
研究 $\mathrm{GL}(n,q)$ 不可约特征标在固定矩阵上取值被 $d$ 整除的比例，证明了当 $n$ 趋于无穷时，若 $q$ 与 $d$ 互素则比例趋于 $1$，不互素且 $g=1$ 时上界为 $\frac{1}{q}$。
Antichains among Divisor Sums of Divisors
本文研究正整数除数在映射到其因子和函数后，原除数格中反链宽度的保留情况，通过整除偏序集乘积宽度刻画最大值，并利用Engel极限理论及素数方向2和3确定精确方差，给出极限公式与渐近结果。
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