Superabundant

In number theory, a superabundant number is a positive integer that is more abundant than any smaller positive integer relative to its size. The concept extends the idea of abundant numbers, which are numbers for which the sum of their proper divisors exceeds the number itself. Superabundant numbers are defined in terms of their abundance index, a measure that compares the ratio of a number's sum of proper divisors to its size, adjusted for its position in the sequence of positive integers.

A number *n* is superabundant if its abundance index is greater than that of every smaller positive integer. The abundance index of a number *n* is calculated as the sum of its proper divisors divided by *n* multiplied by the natural logarithm of *n*. This adjustment accounts for the fact that larger numbers naturally have higher sums of divisors, making direct comparison unfair without normalization.

The sequence of superabundant numbers begins with 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, and so on. These numbers are notable for their strong abundance properties and appear frequently in advanced number-theoretic studies. Superabundant numbers are also related to highly composite numbers, which are numbers with more divisors than any smaller positive integer, though the two concepts are distinct.

Superabundant numbers have applications in various areas of mathematics, including the study of multiplicative functions and the distribution of prime numbers. They are also used in computational number theory to analyze the efficiency of algorithms involving divisors. The sequence continues indefinitely, though identifying superabundant numbers becomes increasingly challenging as their values grow larger.