Piotr Miska

activity

20142025

most citedThe

r

-derangement numbers

23 citations · 50 across the 24 of their papers we have counts for

collaborators

26 papers

math.NT2025

On the arithmetic of multidimensional continued fractions

Piotr Miska, Nadir Murru, Giuliano Romeo

The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by se…

math.NT2024

Binary sequences meet the Fibonacci sequence

Piotr Miska, Bartosz Sobolewski, Maciej Ulas

We introduce a new family of meta-Fibonacci sequences $(f (n))_{n \in N}$ , governed by the recurrence relation $f (n) = a f (n - u_{n} - 1) + b f (n - u_{n} - 2),$ where $\mathbf{u}=(u_{n})…

math.CO2024

Connections between certain numbers related to derangements and $r$ -permutations

Piotr Miska, Błażej Żmija

For non-negative integer parameters $r, u, m, n$ define \begin{align*} \cal{D}(r,u,m,n) := \big\{\ σ\in \cal{S}_{r+n}\ \big|\ σ(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \t…

math.NT2024

On practical sets and $A$ -practical numbers

Andrzej Kukla, Piotr Miska

Let $A$ be a set of positive integers. We define a positive integer $n$ as an $A$ -practical number if every positive integer from the set $\left\{1,\ldots ,\sum_{d\in A, d\mid n}d\…

math.NT2023

On general approach to Bessenrodt-Ono type inequalities and log-concavity property

Krystian Gajdzica, Piotr Miska, Maciej Ulas

In recent literature concerning integer partitions one can find many results related to both the Bessenrodt-Ono type inequalities and log-concavity property. In this note we offer…

math.NT2023

On alternative definition of Lucas atoms and their $p$ -adic valuations

Gessica Alecci, Piotr Miska, Nadir Murru +1

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, wh…

math.NT2022★ 3 cited

On continued fraction partial quotients of square roots of primes

Vítězslav Kala, Piotr Miska

We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sq…

math.NT2022

Characteristics of distributions of sets and their $(R)$ - and $(N)$ -denseness

Piotr Miska, János T. Tóth

Let $0 \leq q \leq 1$ and $N$ denotes the set of all positive integers. In this paper we will deal with it too the family $U (x^{q})$ of all regularly distributed set…

math.NT2022

On Frobenius problem with restrictions on common divisors of coefficients

Piotr Miska, Maciej Zakarczemny

Let $m, s, t$ are positive integers with $t \leq s - 2$ and $a_{1}, a_{2}, \dots, a_{s}$ are positive integers such that $(a_{1}, a_{2}, \dots, a_{s - 1}) = 1$ . In the paper we prove that every sufficien…

math.AC2022★ 2 cited

On Waring numbers of henselian rings

Tomasz Kowalczyk, Piotr Miska

Let $n > 1$ be a positive integer. Let $R$ be a henselian local ring with residue field $k$ of $n$ th level $s_{n} (k)$ . We give some upper and lower bounds for the $n$ th Waring number $…

math.NT2022

On the Diophantine equation $σ_{2} (X_{n}) = σ_{n} (X_{n})$

Piotr Miska, Maciej Ulas

In this note we investigate the set $S (n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of sol…

math.NT2022

Binomial coefficients, roots of unity and powers of prime numbers

Piotr Miska

Let $t \in N_{+}$ be given. In this article we are interested in characterizing those $d \in N_{+}$ such that the congruence $$\frac{1}{t}\sum_{s=0}^{t-1}{n+dζ_t^s\choose…

math.NT2022

$p$ -Adic quotient sets: diagonal forms

Deepa Antony, Rupam Barman, Piotr Miska

For a set of integers $A$ , we consider $R (A) = {a / b : a, b \in A, b \neq = 0}$ . It is an open problem to study the denseness of $R (A)$ in the $p$ -adic numbers when $A$ is the set of non…

math.NT2021★ 4 cited

Stirling number and periodic points

Piotr Miska, Tom Ward

We introduce the notion of almost realizability, an arithmetic generalization of realizability for integer sequences, which is the property of counting periodic points for some map…

math.NT2020

A note on $p$ -adic denseness of quotients of values of quadratic forms

Piotr Miska

Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for in…

math.NT2020

On two conjectures regarding generalized sequence of derangements

Eryk Lipka, Piotr Miska

The second author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures…

math.NT2019

On distribution of subsequences of primes having prime indices with respect to the $(R)$ -denseness and convergence exponent

Piotr Miska, János T. Tóth, Błażej Żmija

Denote by $N$ and $P$ the set of all positive integers and prime numbers, respectively. Let $P = {p_{1} < p_{2} < \dots < p_{n} < \dots}$ , where $p_{n}$ is the $n$ -th p…

math.NT2019

On consecutive 1's in continued fractions expansions of square roots of prime numbers

Piotr Miska, Maciej Ulas

In this note, we study the problem of existence of sequences of consecutive 1's in the periodic part of the continued fractions expansions of square roots of primes. We prove uncon…

math.NT2019

On some properties of the number of permutations being products of pairwise disjoint $d$ -cycles

Piotr Miska, Maciej Ulas

Let $d \geq 2$ be an integer. In this paper we study arithmetic properties of the sequence $(H_{d} (n))_{n \in N}$ , where $H_{d} (n)$ is the number of permutations in $S_{n}$ being produ…

math.NT2018★ 6 cited

$p$ -adic denseness of members of partitions of $N$ and their ratio sets

Piotr Miska, Carlo Sanna

The ratio set of a set of positive integers $A$ is defined as $R (A) := {a / b : a, b \in A}$ . The study of the denseness of $R (A)$ in the set of positive real numbers is a classi…

math.NT2018★ 12 cited

On the p-adic denseness of the quotient set of a polynomial image

Piotr Miska, Nadir Murru, Carlo Sanna

The quotient set, or ratio set, of a set of integers $A$ is defined as $R (A) := {a / b : a, b \in A, b \neq = 0}$ . We consider the case in which $A$ is the image of $\mat…

math.NT2018

A note on $p$ -adic locally analytic functions with application to behavior of the $p$ -adic valuations of Stirling numbers

Piotr Miska

The aim of this paper is to prove conjectures concerning $p$ -adic valuations of Stirling numbers of the second kind $S (n, k)$ , $n, k \in N_{+}$ , stated by Amdeberhan, Manna and…

math.NT2018★ 23 cited

The $r$ -derangement numbers

Chenying Wang, Piotr Miska, István Mező

The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are res…

math.NT2017

Arithmetic properties of coefficients of power series expansion of $\prod_{n = 0}^{\infty} (1 - x^{2^{n}})^{t}$ (with an Appendix by Andrzej Schinzel)

Maciej Gawron, Piotr Miska, Maciej Ulas

Let $F (x) = \prod_{n = 0}^{\infty} (1 - x^{2^{n}})$ be the generating function for the Prouhet-Thue-Morse sequence $((- 1)^{s_{2} (n)})_{n \in N}$ . In this paper we initiate the study of the…

On the arithmetic of multidimensional continued fractions

Binary sequences meet the Fibonacci sequence

Connections between certain numbers related to derangements and r-permutations

On practical sets and A-practical numbers

On general approach to Bessenrodt-Ono type inequalities and log-concavity property

On alternative definition of Lucas atoms and their p-adic valuations

On continued fraction partial quotients of square roots of primes

Characteristics of distributions of sets and their (R)- and (N)-denseness

On Frobenius problem with restrictions on common divisors of coefficients

On Waring numbers of henselian rings

On the Diophantine equation σ2​(Xn​)=σn​(Xn​)

Binomial coefficients, roots of unity and powers of prime numbers

p-Adic quotient sets: diagonal forms

Stirling number and periodic points

A note on p-adic denseness of quotients of values of quadratic forms

On two conjectures regarding generalized sequence of derangements

On distribution of subsequences of primes having prime indices with respect to the (R)-denseness and convergence exponent

On consecutive 1's in continued fractions expansions of square roots of prime numbers

On some properties of the number of permutations being products of pairwise disjoint d-cycles

p-adic denseness of members of partitions of N and their ratio sets

On the p-adic denseness of the quotient set of a polynomial image

A note on p-adic locally analytic functions with application to behavior of the p-adic valuations of Stirling numbers

The r-derangement numbers

Arithmetic properties of coefficients of power series expansion of ∏n=0∞​(1−x2n)t (with an Appendix by Andrzej Schinzel)

Connections between certain numbers related to derangements and $r$ -permutations

On practical sets and $A$ -practical numbers

On alternative definition of Lucas atoms and their $p$ -adic valuations

Characteristics of distributions of sets and their $(R)$ - and $(N)$ -denseness

On the Diophantine equation $σ_{2} (X_{n}) = σ_{n} (X_{n})$

$p$ -Adic quotient sets: diagonal forms

A note on $p$ -adic denseness of quotients of values of quadratic forms

On distribution of subsequences of primes having prime indices with respect to the $(R)$ -denseness and convergence exponent

On some properties of the number of permutations being products of pairwise disjoint $d$ -cycles

$p$ -adic denseness of members of partitions of $N$ and their ratio sets

A note on $p$ -adic locally analytic functions with application to behavior of the $p$ -adic valuations of Stirling numbers

The $r$ -derangement numbers

Arithmetic properties of coefficients of power series expansion of $\prod_{n = 0}^{\infty} (1 - x^{2^{n}})^{t}$ (with an Appendix by Andrzej Schinzel)