Related papers: The Carmichael numbers up to $10^{16}$
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as…
We consider a multinomial distribution, where the number of cells increases and the cell-probabilities decreases as the number of observations grows. The probabilities of large deviations of statistics, which has form of a sum of Borel…
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit…
We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.
We introduce $p$-derivations and give a few basic ways in which they act like derivatives by numbers.
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals.…
We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents…
Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.
We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…
In this paper, we give some interesting identities of poly-Cauchy numbers and polynomials arising from umbral calculus.
How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the \emph{rearrangement number}, a new cardinal…
Multiple analogues of certain families of combinatorial numbers are recently constructed by the author in terms of well poised Macdonald functions, and some of their fundamental properties are developed. In this paper, we present…
In this paper, we study the Carlitz's degenerate Bernoulli numbers and polynomials and give some formulae and identities related to those numbers and polynomials.
In this paper, we introduce multi-Lah numbers and multi-Stirling numbers of the first kind and recall multi-Bernoulli numbers, all of whose generating functions are given with the help of multiple logarithm. The aim of this paper is to…
In this article we prove that equation $\phi(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product…
A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…
The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.
In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.
In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.