Highly composite number


A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d > d for all n < N. For example, 6 is highly composite because d = 4, and for n = 1,2,3,4,5, you get d = 1,2,2,3,2, respectively, which are all less than 4.
A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers are not actually composite numbers; however, all further terms are.
Ramanujan wrote a paper on highly composite numbers in 1915.
The mathematician Jean-Pierre Kahane suggested that Plato must have known about highly composite numbers as he deliberately chose such a number, 5040, as the ideal number of citizens in a city. Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040.

Examples

The first 41 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled d. Asterisks indicate superior highly composite numbers.
OrderHCN
n		prime
factorization		prime
exponents		number
of prime
factors		primorial
factorization
1101
22*112
34223
46*1,124
512*2,136
6243,148
7362,249
8484,1510
960*2,1,1412
10120*3,1,1516
111802,2,1518
122404,1,1620
13360*3,2,1624
147204,2,1730
158403,1,1,1632
1612602,2,1,1636
1716804,1,1,1740
182520*3,2,1,1748
195040*4,2,1,1860
2075603,3,1,1864
21100805,2,1,1972
22151204,3,1,1980
23201606,2,1,11084
24252004,2,2,1990
25277203,2,1,1,1896
26453604,4,1,110100
27504005,2,2,110108
2855440*4,2,1,1,19120
29831603,3,1,1,19128
301108805,2,1,1,110144
311663204,3,1,1,110160
322217606,2,1,1,111168
332772004,2,2,1,110180
343326405,3,1,1,111192
354989604,4,1,1,111200
365544005,2,2,1,111216
376652806,3,1,1,112224
38720720*4,2,1,1,1,110240
3910810803,3,1,1,1,110256
401441440*5,2,1,1,1,111288
4121621604,3,1,1,1,111320

The divisors of the first 20 highly composite numbers are shown below.
nDivisors of n
111
221, 2
431, 2, 4
641, 2, 3, 6
1261, 2, 3, 4, 6, 12
2481, 2, 3, 4, 6, 8, 12, 24
3691, 2, 3, 4, 6, 9, 12, 18, 36
48101, 2, 3, 4, 6, 8, 12, 16, 24, 48
60121, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
120161, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
180181, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
240201, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240
360241, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
720301, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720
840321, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840
1260361, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260
1680401, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680
2520481, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520
5040601, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040
7560641, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 54, 56, 60, 63, 70, 72, 84, 90, 105, 108, 120, 126, 135, 140, 168, 180, 189, 210, 216, 252, 270, 280, 315, 360, 378, 420, 504, 540, 630, 756, 840, 945, 1080, 1260, 1512, 1890, 2520, 3780, 7560

The table below shows all 72 divisors of 10080 by writing it as a product of two numbers in 36 different ways.
The 15,000-th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:
where is the th successive prime number, and all omitted terms are factors with exponent equal to one. More concisely, it is the product of seven distinct primorials:
where is the primorial.

Prime factorization

Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization:
where are prime, and the exponents are positive integers.
Any factor of n must have the same or lesser multiplicity in each prime:
So the number of divisors of n is:
Hence, for a highly composite number n,
  • the k given prime numbers pi must be precisely the first k prime numbers ; if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors ;
  • the sequence of exponents must be non-increasing, that is ; otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors.
Also, except in two special cases n = 4 and n = 36, the last exponent ck must equal 1. It means that 1, 4, and 36 are the only square highly composite numbers. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials or, alternatively, the smallest number for its prime signature.
Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example, 96 = 25 × 3 satisfies the above conditions and has 12 divisors but is not highly composite since there is a smaller number which has the same number of divisors.

Asymptotic growth and density

If Q denotes the number of highly composite numbers less than or equal to x, then there are two constants a and b, both greater than 1, such that
The first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988. We have
and