A highly abundant number has a larger sum of divisors \(\sigma(n)\) than any smaller positive integer — the σ-analogue of the [[highly-composite]] numbers (which maximise the divisor count instead). The sequence begins 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48.
They were introduced by Leonidas Alaoglu and Paul Erdős in 1944. Every [[perfect-number]] and every [[highly-composite]] number beyond a small threshold is highly abundant, and the two families overlap heavily without one containing the other. NumberWiki finds them with a sum-of-divisors sieve up to one million.