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superabundant numbers
A number  $n$  is called superabundant if  $\sigma(n)/n > \sigma(m)/m$  for all the numbers  $m<n$, i.e., if it sets a new record in the ratio  $\sigma(n)/n$, which is called abundancy index.

The first 19 superabundant numbers coincide with the first 19 highly composite numbers. The smallest superabundant number which is not highly composite is 1163962800.

The first superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of superabundant numbers

A graph displaying how many superabundant numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of superabundant numbers
Useful links Mathworld, Superabundant Number
Wikipedia, Superabundant number
OEIS, Sequence A004394
Superabundant numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 24 aban 12 24 36 48 60 120 180 240 360 720 840 abundant 12 24 36 48 60 120 180 240 360 720 + 8648640 10810800 21621600 36756720 admirable 12 24 120 alternating 12 36 amenable 12 24 36 48 60 120 180 240 360 720 + 183783600 367567200 698377680 735134400 apocalyptic 720 840 10080 15120 25200 27720 arithmetic 60 840 2520 10080 15120 27720 110880 166320 332640 554400 + 2162160 4324320 7207200 8648640 betrothed 48 binomial 36 120 25200 27720 720720 2162160 7207200 compositorial 24 congruent 24 60 120 180 240 720 840 2520 10080 25200 + 1441440 2162160 4324320 8648640 constructible 12 24 48 60 120 240 Cunningham 24 48 120 360 840 1680 5040 d-powerful 24 dig.balanced 12 120 180 240 2520 4324320 7207200 8648640 10810800 double fact. 48 droll 240 Duffinian 36 eban 36 60 eRAP 24 esthetic 12 Eulerian 120 evil 12 24 36 48 60 120 180 240 360 720 + 183783600 367567200 698377680 735134400 factorial 24 120 720 5040 Friedman 1260 gapful 120 180 240 360 1260 1680 2520 10080 15120 25200 + 27935107200 41902660800 48886437600 80313433200 happy 15120 55440 110880 166320 554400 2162160 harmonic 332640 Harshad 12 24 36 48 60 120 180 240 360 720 + 2327925600 2793510720 3491888400 6983776800 hexagonal 120 2162160 highly composite 12 24 36 48 60 120 180 240 360 720 + 195643523275200 288807105787200 577614211574400 866421317361600 iban 12 24 120 240 720 27720 277200 720720 idoneal 12 24 48 60 120 240 840 inconsummate 840 332640 665280 interprime 12 60 120 180 240 55440 110880 554400 73513440 Jordan-Polya 12 24 36 48 120 240 720 5040 10080 junction 27720 166320 720720 4324320 8648640 katadrome 60 720 840 lonely 120 Lynch-Bell 12 24 36 48 magnanimous 12 metadrome 12 24 36 48 nialpdrome 60 720 840 55440 554400 nonagonal 24 nude 12 24 36 48 O'Halloran 12 36 60 oban 12 36 60 360 720 octagonal 1680 odious 166320 332640 665280 panconsummate 12 24 36 pandigital 120 180 pentagonal 12 pernicious 12 24 36 48 166320 332640 665280 Perrin 12 plaindrome 12 24 36 48 power 36 powerful 36 practical 12 24 36 48 60 120 180 240 360 720 + 3603600 4324320 7207200 8648640 prim.abundant 12 pronic 12 240 1260 4324320 pseudoperfect 12 24 36 48 60 120 180 240 360 720 + 332640 554400 665280 720720 Ruth-Aaron 24 Saint-Exupery 60 square 36 straight-line 840 super Niven 12 24 36 48 60 120 240 360 840 2520 + 8648640 61261200 122522400 1102701600 super-d 15120 332640 tau 12 24 36 60 180 240 360 720 1260 1680 + 3603600 8648640 61261200 735134400 tetrahedral 120 27720 7207200 triangular 36 120 25200 2162160 tribonacci 24 trimorphic 24 uban 12 36 48 60 Ulam 36 48 180 720 110880 7207200 unprimeable 840 1260 1680 5040 10080 25200 27720 166320 277200 554400 + 2162160 3603600 7207200 8648640 untouchable 120 1680 15120 25200 27720 277200 332640 720720 vampire 1260 wasteful 12 24 36 48 60 120 180 240 360 720 + 3603600 4324320 7207200 8648640 Zuckerman 12 24 36 Zumkeller 12 24 48 60 120 180 240 360 720 840 + 15120 25200 27720 55440 zygodrome 554400