Related papers: Asymptotic density of Motzkin numbers modulo small…

We establish the asymptotic density and some structure results for the Motzkin numbers modulo primes up to 29 using automata.

We establish a lower bound of 2/p(p-1) for the asymptotic density of the Motzkin numbers divisible by a general prime number p > 3. We provide a criteria for when this asymptotic density is actually 1. We also provide a partial…

We establish the asymptotic density of the Catalan numbers modulo 3 and modulo powers of 2.

The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in…

This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums…

We prove that the average of the $k$-th smallest prime quadratic non-residue modulo a prime approximates the $2k$-th smallest prime.

We give an asymptotic formula for the number of non-zero coefficients of modular forms (mod p).

Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$…

In this paper we determine the asymptotic density of coprime fractions in those of the reduced fractions of number fields. When ordered by norms of denominators, we count a fraction as soon as it ``appears'' for the first time and no later.…

We provide an elementary proof of an asymptotic formula for prime counting functions. As a minor application we give a new reduction of the proof of Chebotar\"ev's density theorem to the cyclic case.

We give a formula for the density of $0$ in the sequence of generalized Motzkin numbers, $M^{a, b}_n$, modulo a prime, $p$, in terms of the first $p$ generalized central trinomial coefficients $T^{a, b}_n\bmod p$ (with $n<p$). We apply our…

We consider several problems about pseudoprimes. First, we look at the issue of their distribution in residue classes. There is a literature on this topic in the case that the residue class is coprime to the modulus. Here we provide some…

We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such…

Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we…

Precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite- and infinite-dimensional settings. Our results are based on the existence of dominating points for these sets, a related…

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k…

We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively (see J. Arves\'u et al. J. Comput. Appl. Math., 153, (2003)). We use an algebraic function formulation for the solution of the…

We determine the asymptotic density $\delta_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$,…

The density of a subgroupoid with respect to a free groupoid is defined as the asymptotic ratio of their growths. This notion can be interpreted as a generalisation of the index's inverse for groups or as the probability of an element…