Related papers: The 2-adic valuations of Stirling numbers of the s…

The Stern sequence (s(n)) is defined by s(0) = 0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1). Stern showed in 1858 that gcd(s(n),s(n+1)) = 1, and that for every pair of relatively prime positive integers (a,b), there exists a unique n…

Given a set $S$ consisting of $n$ points in $\mathbb{R}^d$ and one or two vantage points, we study the number of orderings of $S$ induced by measuring the distance (for one vantage point) or the average distance (for two vantage points)…

The ordinary factorial may be written in terms of the Stirling numbers of the second kind as shown by Quaintance and Gould and the odd double factorial in terms of the Stirling numbers of the first kind as shown by Callan. During the…

We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the…

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2.…

Let $\A=\{a_1<a_2<a_3.....<a_n<...\}$ be an infinite sequence of integers and let $R_2(n)=|\{(i,j):\ \ a_i+a_j=n;\ \ a_i,a_j\in \A;\ \ i\le j\}|$. We define $S_k=\s_{l=1}^k(R_2(2l)-R_2(2l+1))$. We prove that, if $L^{\infty}$ norm of…

Congruences modulo prime powers involving generalized Harmonic numbers are known. While looking for similar congruences, we have encountered a curious triangular array of numbers indexed with positive integers $n,k$, involving the Bernoulli…

Fix n>2. Let s be a principally embedded sl(2)-subalgebra in sl(n). A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite dimensional…

In Combinatorics Stirling numbers may be defined in several ways. One such definition is given in [1], where an extensive consideration of Stirling numbers is presented. In this paper an alternative definition of Stirling numbers of both…

Two doubly indexed families of homogeneous and isobaric polynomials in several indeterminates are considered: the (partial) exponential Bell polynomials $B_{n,k}$ and a new family $S_{n,k}$ such that $X_1^{-(2n-1)}S_{n,k}$ and $B_{n,k}$…

It is known that the Scholz conjecture on addition chains is true for all integers $n$ with $\ell(2n) = \ell(n)+1$. There exists infinitely many integers with $\ell(2n) \leq \ell(n)$ and we don't know if the conjecture still holds for them.…

Given positive integers $n,k$ with $k\leq n$, we consider the number of ways of choosing $k$ subsets of $\{1,\ldots,n\}$ in such a way that the union of these subsets gives $\{1,\ldots,n\}$ and they are not subsets of each other. We refer…

We introduce a new sequence of unsigned degenerate Stirling numbers of the first kind. Following the work of Adell-Lekuona, who represented unsigned Stirling numbers of the first kind as multiples of the expectations of specific random…

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/k$ be the $n$th harmonic number. Z. W. Sun conjectured that for any prime $p\ge 5$, $$ \sum_{k=1}^{p-1}\frac{H_k}{k\cdot 2^k} \equiv7/24pB_{p-3}\pmod{p^2}. $$ This conjecture is recently…

In 2022, Broudy and Lovejoy extensively studied the function $S(n)$ which counts the number of overpartitions of \emph{Schur-type}. In particular, they proved a number of congruences satisfied by $S(n)$ modulo $2$, $4$, and $5$. In this…

In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course…

Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions,…

The 2-adic valuation of an integer n which is the exponent of the highest power of 2 that divides n. In this paper, we give representations of certain restricted partition functions in terms of 2-adic valuation.

Given an integer g, g > 1, an integer w, -1 < w <g - 1, and a set of g distinct numbers, c_1, ..., c_g, we present a conjectured identity for Stirling numbers of the first kind. We have proven all the equalities in case g < 7; and for the…

We determine all pairs of positive integers $(a,b)$ such that $a+b$ and $a \times b$ have the same decimal digits in reverse order: \[ (2,2), (9,9), (3,24), (2,47), (2,497), (2,4997), (2,49997), \ldots \] We use deterministic finite…