Related papers: On $k$-regularity of sequences of valuations and l…

For integers $k \geq 2$ and $n \neq 0$, let $v_k(n)$ denotes the greatest nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $u_n$ be a nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} = a u_{n +…

In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base $b$. For the sequence given by the last digits of $n^n$ in base $b$, we prove its periodicity using different techniques than those used…

We show that the $p$-adic valuation of the sequence of Fibonacci numbers is a $p$-regular sequence for every prime $p$. For $p \neq 2, 5$, we determine that the rank of this sequence is $\alpha(p) + 1$, where $\alpha(m)$ is the restricted…

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a…

We generalize results on the $p$-adic valuations of $S(n,k)$, the Stirling number of the second kind and $s(n,k)$ the Stirling number of the first kind. We have several new estimates for these valuations, along with criteria for when the…

The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number.…

We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer…

We prove that the sequence of the last nonzero digits of factorials in every integer base $b>2$ is not eventually periodic. We also extend the Adamczewski--Bugeaud criterion, originally formulated for integer base expansions, to Cantor base…

We examine the behavior of the sequences of $p$-adic valuations of quadratic polynomials with integer coefficients for an odd prime $p$ through tree representations. Under this representation, a finite tree corresponds to a periodic…

In this paper, we extend the $p$-adic valuations of the M\"obius duals of Lucas sequences, originally obtained by Carmichael for regular Lucas sequences to irregular Lucas sequences. We conclude with a brief observation about the…

In this paper, we extend the $p$-adic valuations originally obtained by Carmichael for the sequences obtained by applying M\"obius inversion to Lucas sequences to $p$-adic congruences, from which we immediately derive corresponding…

We prove that many sequences of positive numbers $(a_n)$ defined by finite linear difference equations $a_{n+k}=c_{k-1}a_{n+k-1}+...+c_0a_n$ with suitable non negative reals coefficients $c_i$ satisfy Bendford's Law on the first digit in…

Let $p$ be a prime. Let $A$ and $B$, $A \ge B \ge 0$, be integers with base $p$ expansions $A = \alpha_i\alpha_{i-1}\dots \alpha_0$ and $B = \beta_i\beta_{i-1}\dots \beta_0$. Lucas proved that $$\binom{A}{B} \equiv…

For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $\nu_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $\nu_{p}(Q(n))$ is shown to be either periodic or unbounded. The first…

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Ap\'ery-like recurrence relation: these include Ap\'ery, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and…

We define the Narayana sequence $\{a_n\}_{n\geq 0}$ as the one satisfying the linear recurrence relation $a_n = a_{n-1}+a_{n-3}$ for $n\geq 3$, with initial values $a_0 = 0$ and $a_1 = a_2=1$. In this paper, we fully characterize the…

We say that an arithmetical function $S:\mathbb{N}\rightarrow\mathbb{Z}$ has Lucas property if for any prime $p$, \begin{equation*} S(n)\equiv S(n_{0})S(n_{1})\ldots S(n_{r})\pmod p, \end{equation*} where $n=\sum_{i=0}^{r}n_{i}p^{i}$, with…

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in \cite{ST}. The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually…

In number theory, we know Legendre's formula $ v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$.…

For a positive integer $k$, let \[ \sigma_k(n)=\sum_{d\mid n} d^k \] be the divisor function of order $k$, and let $\nu_p(m)$ denote the $p$-adic valuation of an integer $m$. Motivated by recent work on the $p$-adic valuation of…