The Biharmonic mean

TL;DR

This paper introduces the biharmonic mean and biharmonic numbers, revealing their properties, divisibility features, and potential to characterize prime and semi-prime numbers through connections with linear recurrence sequences.

Contribution

It proposes the biharmonic mean as a new mathematical concept and explores its properties, including a novel characterization of primes and semi-primes.

Findings

Introduction of the biharmonic mean and numbers

Divisibility properties of biharmonic numbers

Characterization of semi-prime biharmonic numbers

Abstract

We briefly describe some well-known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we call the biharmonic mean. The biharmonic mean allows to introduce the biharmonic numbers, providing a new characterization for primes. Moreover, we highlight some interesting divisibility properties and we characterize the semi--prime biharmonic numbers showing their relationship with linear recurrent sequences that solve certain Diophantine equations.