On odd integers and their couples of divisors

Abstract

A composite odd integer can be expressed as product of two odd integers. Possibly\xa0this decomposition is not unique. From 2n + 1 = (2i + 1)(2j + 1) it follows that\xa0n = i + j + 2ij. This form of n characterizes the composite odd integers. It allows\xa0the formulation of simple algorithms to compute all the couples of divisors of odd\xa0integers and to identify the odd inetegers with the same number of couples of divisors\xa0(including the primes, with the number of non trivial divisors equal to zero). The\xa0distributions of odd integers ≤ 2n+1 vs. the number of their couples of divisors have\xa0been computed up to n ≃ 5 10^7, and the main features are illustrated.