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highly composite numbers
A number  $n$  is called highly composite if it has more divisors than any smaller number, i.e., if it sets a new record in the number of divisors.

For example, 60, it is highly composite because it has 12 divisors and none of the numbers up to 59 has 12 or more divisors.

The first 19 highly composite numbers coincide with the first 19 superabundant numbers, and the first one which is not superabundant is 7560.

All the highly composite numbers greater than 1 have a prime factorization  $2^{e_2}\cdot3^{e_3}\cdots p^{e_p}$  where all the primes from 2 to  $p$  appear. Moreover, the exponents are nonincreasing, i.e.,  $e_2\ge e_3\ge\cdots\ge e_p$  and the last one ( $e_p$) is always 1 except for the numbers  $4=2^2$  and  $36=2^2\cdot3^2$.

The first highly composite numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080 more terms

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

A graph displaying how many highly composite numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of highly composite numbers
Useful links Mathworld, Highly Composite Number
Wikipedia, Highly composite number
OEIS, Sequence A002182
Highly composite 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 + 21621600 32432400 36756720 43243200 admirable 12 24 120 alternating 12 36 amenable 12 24 36 48 60 120 180 240 + 367567200 551350800 698377680 735134400 apocalyptic 720 840 7560 10080 15120 20160 25200 27720 arithmetic 60 840 2520 7560 10080 15120 27720 83160 + 2882880 4324320 7207200 8648640 betrothed 48 binomial 36 120 25200 27720 720720 2162160 7207200 compositorial 24 congruent 24 60 120 180 240 720 840 2520 + 2882880 4324320 6486480 8648640 constructible 12 24 48 60 120 240 Cunningham 24 48 120 360 840 1680 5040 17297280 d-powerful 24 dig.balanced 12 120 180 240 2520 4324320 6486480 7207200 8648640 10810800 14414400 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 + 367567200 551350800 698377680 735134400 factorial 24 120 720 5040 Friedman 1260 45360 498960 gapful 120 180 240 360 1260 1680 2520 7560 + 64250746560 73329656400 80313433200 97772875200 happy 15120 45360 55440 110880 166320 221760 554400 1081080 2162160 harmonic 332640 Harshad 12 24 36 48 60 120 180 240 + 3491888400 4655851200 5587021440 6983776800 heptagonal 45360 hexagonal 120 2162160 iban 12 24 120 240 720 27720 277200 720720 idoneal 12 24 48 60 120 240 840 inconsummate 840 20160 45360 332640 665280 interprime 12 60 120 180 240 7560 55440 110880 + 554400 2882880 17297280 73513440 Jordan-Polya 12 24 36 48 120 240 720 5040 10080 20160 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 83160 166320 332640 498960 665280 panconsummate 12 24 36 pandigital 120 180 pentagonal 12 pernicious 12 24 36 48 83160 166320 332640 665280 Perrin 12 plaindrome 12 24 36 48 power 36 powerful 36 practical 12 24 36 48 60 120 180 240 + 4324320 6486480 7207200 8648640 prim.abundant 12 pronic 12 240 1260 50400 4324320 pseudoperfect 12 24 36 48 60 120 180 240 + 498960 554400 665280 720720 Ruth-Aaron 24 Saint-Exupery 60 Smith 2882880 square 36 straight-line 840 super Niven 12 24 36 48 60 120 240 360 + 110270160 122522400 1102701600 2205403200 super-d 15120 332640 498960 superabundant 12 24 36 48 60 120 180 240 + 195643523275200 288807105787200 577614211574400 866421317361600 tau 12 24 36 60 180 240 360 720 + 61261200 245044800 551350800 735134400 tetrahedral 120 27720 7207200 triangular 36 120 25200 2162160 tribonacci 24 trimorphic 24 uban 12 36 48 60 Ulam 36 48 180 720 50400 110880 6486480 7207200 unprimeable 840 1260 1680 5040 10080 25200 27720 50400 + 3603600 6486480 7207200 8648640 untouchable 120 1680 15120 20160 25200 27720 45360 50400 + 277200 332640 498960 720720 vampire 1260 wasteful 12 24 36 48 60 120 180 240 + 4324320 6486480 7207200 8648640 Zuckerman 12 24 36 Zumkeller 12 24 48 60 120 180 240 360 + 45360 50400 55440 83160 zygodrome 554400