Related papers: On the primes in floor function sets

This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting…

The purpose of this note is to report on the discovery of the primes of the form $p=1+n!\sum n$, for some natural numbers $n>0$. The number of digits in the prime p are approximately equal to $\lfloor log_{10}(1+n!\sum n)\rceil+1$.

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number…

We prove some properties of the sequence $\{a_n\}_{n\ge1}$ defined by $a_n=\pi(n)-\pi\bigl(\textstyle\sum_{k=1}^{n-1}a_k\bigr).$

Fix $\delta\in(0,1]$, $\sigma_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $\pi_{\mathcal P}(x)$ satisfies an…

Definition of the number of prime numbers in the given interval

The sequence of the primes $p$ for which a variety over $\mathbb{Q}$ has no $p$-adic point plays a fundamental role in arithmetic geometry. This sequence is deterministic, however, we prove that if we choose a typical variety from a family…

Classical parking functions are a generalization of permutations that appear in many combinatorial structures. Prime parking functions are indecomposable components such that any classical parking function can be uniquely described as a…

Let $n$ be a positive integer and let $A$ be nonempty finite set of positive integers. We say that $A$ is relatively prime if $\gcd(A) =1$ and that $A$ is relatively prime to $n$ if $\gcd(A,n)=1$. In this work we count the number of…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes

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…

V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…

Let A be a nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Suppose P is a set of primes, such that for every p in P, every prime factor of p-1 is also in P. If P does not contain all primes, we apply a new sieve method to show that the counting function of P is O(x^{1-c}) for some c>0, where c…

Let $F_p$ be the field of a prime order $p$. Then for any positive integer $n>1$, for any $\epsilon>0$, and for any subset $A$ of $F_p$, every element of $F_p$ can be represented as a sum of $N$ elements, each of them is a product of $n$…

The prime-counting function $\pi(x)$ which returns the number of primes smaller or equal to a given number is a topic of interest in number theory. An algorithm based on a cyclic group isomorphic to $Z/nZ$, the so-called $Z$-functions, was…

A sharp asymptotic formula for the sum of reciprocals of $\pi(n)$ is derived, where $\pi(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivi\'c and L. Panaitopol.

Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the…