Computational and Statistical Study of the Relationship Between σ(n) and φ(n) in Natural Numbers

Abstract

The focus of this work is to understand the classification of natural numbers, by implementing a computational approach using the Sum of Divisors Function by applying MATLAB. Numbers that are classified into three broad groups: Deficient Numbers, Perfect Numbers and Abundant Numbers based on the relationship between the sum of their Positive Divisors and twice the number. The first aim of this work is to break down the frequency of these classes into various groups in numerical ranges and to study the behaviour of these classes as the range increases. A sum of divisors algorithm in MATLAB was developed to compute the sum of divisors of each natural number and classify it as an either abundant or deficient number. Several runs totaling up to 100,000 were performed using the system. The outcome exhibited that the number class of Deficient numbers are found in larger numbers then the numbers that are Abundant, which is about 1-quarter of the numbers. The numbers that were found to be perfect were found to be very few, being only four in the range tested. The results reveal a statistically stable behavior of the ratio σ(n)/φ(n) across different numerical ranges, suggesting an underlying invariance in the interaction between divisor-based and multiplicative arithmetic functions.