A positive integer is called a (Ore’s) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than 1 is even. If Ore’s conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than 1 must be divisible by a prime greater than 105.

A natural number n is said to be harmonic when the harmonic mean H(n) of its positive divisors is an integer. These were first introduced almost fifty years ago. In this paper, all harmonic numbers less than 2 × 109 are listed, along with some other useful tables, and all harmonic numbers n with H(n) ≤ 13 are determined.

Harmonic maps between Riemannian manifolds are maps which extremize a certain natural energy functional; they appear in particle physics as nonlinear sigma models. Their infinitesimal deformations are called Jacobi fields. It is important to know whether the Jacobi fields along the harmonic maps between given Riemannian manifolds are integrable, i.e., arise from genuine variations through harmo...

A comparative analysis between the fundamental and second cyclotron harmonics of gyrotron backward-wave oscillators (gyro-BWOs) is presented. The simulation results reveal that nonlinear field contraction is a common feature for both harmonic interactions. Besides, the electron transit angle, used to characterize the axial modes of the fundamental harmonic TE11 mode at the start-oscillation con...

The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left hand side of the path from a point near the end of the current path. We prove that the harmonic explorer converges to SLE4 as the grid gets finer.

This paper presents an efficient Harmonic-CELP hybrid coder at 2.4 kbps utilizing the well-known characteristics of the Harmonic and CELP coders. According to frame voicing decision, the proposed hybrid coder switches the RP-VSELP coder as a fast CELP in case of unvoiced, or an improved Harmonic coder in case of voiced. The proposed Harmonic-CELP hybrid coder has several features as follows: fa...

This investigation was designed to study the characteristics of a marker for harmonic processing and to test whether it could be used for differentiating harmonic processing capabilities. The ®rst three chords of an ordinary musical cadenca were presented to the left ear to establish a harmonic context followed by a harmonic or non-harmonic target tone. Cadencas were presented rapidly and rando...

The aim oj this article is to study the convergence oj the Euler harmonic series. Firstly, the results concerning the convergence oj the Smaralldache and Erdos harmonic junctions are reviewed Secondly, the Euler harmonic series is proved to be convergent jor a> I, and divergent otherwise. Finally, the slims of the Euler harmonic series are given

We study the stability of harmonic morphisms as a subclass of harmonic maps. As a general result we show that any harmonic morphism to a manifold of dimension at least three is stable with respect to some Riemannian metric on the target. Furthermore we link the index and nullity of the composition of harmonic morphisms with the index and nullity of the composed maps.

Recently, amount of work about the A-harmonic equation for the differential forms has been done. In fact, the A-harmonic equation is an important generalization of the p-harmonic equation in R, p > 1, and the p-harmonic equation is a natural extension of the usual Laplace equation see 1 for the details . The reverse Hölder inequalities have been widely studied and frequently used in analysis an...