Related papers: The denominators of harmonic numbers (Revised)

For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with…

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists…

We consider families of non-self-adjoint perturbations of self-adjoint harmonic and anharmonic oscillators. The norms of spectral projections of these operators are found to grow at intermediate rates from arbitrarily slowly to…

Solutions to Laplace's equation are called harmonic functions. Harmonic functions arise in many applications, such as physics and the theory of stochastic processes. Of interest classically are harmonic polynomials, which have a simple…

Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study…

In the present paper, we have found new upper bounds for chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete critical distance $d$ the chromatic number of…

We study the L^p-distortion of finite quotients of amenable groups. In particular, for every number p larger or equal than 2, we prove that the l^p-distortion of the finite lamplighter group grows like (\log n)^{1/p}. We also give the…

For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and…

The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T^{N,\chi}_{l,k} for all primes l, all weights k>1 and all…

In this paper, we apply the power-partible reduction to study arithmetic properties of sums involving Delannoy numbers $D_k$ and polynomials $D_k(z)$. Let $v\in\bN$ and $p$ be an odd prime. It is proved that, for any…

We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this…

New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…

Let $n_1,\cdots,n_r$ be any finite sequence of integers and let $S$ be the set of all natural numbers $n$ for which there exists a divisor $d(x)=1+\sum_{i=1}^{deg(d)}c_ix^i$ of $x^n-1$ such that $c_i=n_i$ for $1\leq i \leq r$. In this paper…

For a given boundary sequence $a=(a_n)_{n\in\mathbb{Z}}$, we construct harmonic extensions $U,V:\mathbb{Z}\times\ \mathbb{N}\to \mathbb{R}$ that serve as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is…

Let n be a non-null positive integer and $d(n)$ is the number of positive divisors of n, called the divisor function. Of course, $d(n) \leq n$. $d(n) = 1$ if and only if $n = 1$. For $n > 2$ we have $d(n) \geq 2$ and in this paper we try to…

By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $\alpha\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence…

We study the supersymmetric partners of the harmonic oscillator with an infinite potential barrier at the origin and obtain the conditions under which it is possible to add levels to the energy spectrum of these systems. It is found that…

We consider a variant of Dickson lemma, where each entry of a vector can be reseted or incremented by 1 in respect to the previous one. We give an example of non dominating sequence of length $2^{2^{\theta (n)}}$. It perfectly match the…

Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of…