# highly abundant number

An integer $n$ is a highly abundant number if $\sigma(n)>\sigma(m)$ for all $m (with $\sigma$ being the sum of divisors function). The highly abundant numbers less than 100 are 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96 (see A002093 in Sloane’s OEIS). Highly abundant numbers are like highly composite numbers except the definition for the latter uses the divisor function $\tau$ instead of $\sigma$. The highly abundant numbers grow much more slowly than the highly composite numbers.

Though the first eight factorials are highly abundant, not all factorials are highly abundant. Two examples: 360360 is more abudant than 362880; and 3492720, 3538080, 3598560, 3603600 are all more abundant than 3628800.

Title highly abundant number HighlyAbundantNumber 2013-03-22 18:20:55 2013-03-22 18:20:55 CompositeFan (12809) CompositeFan (12809) 4 CompositeFan (12809) Definition msc 11A05