On odd integers and their couples of divisors

Abstract

A composite odd integer can be expressed as product of two odd integers. Possibly this decomposition is not unique. From 2n + 1 = (2i + 1)(2j + 1) it follows that n = i + j + 2ij. This form of n characterizes the composite odd integers. It allows the formulation of simple algorithms to compute all the couples of divisors of odd integers and to identify the odd inetegers with the same number of couples of divisors (including the primes, with the number of non trivial divisors equal to zero). The distributions of odd integers ≤ 2n+1 vs. the number of their couples of divisors have been computed up to n ≃ 5 10^7, and the main features are illustrated.