A highly composite number is a positive integer that has more divisors than all smaller positive integers. If denotes the number of divisors of a positive integer , then a positive integer is highly composite if for all . For example, 6 is highly composite because , and for , you get , 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 (1 and 2) 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 (= 7!), as the ideal number of citizens in a city.. Kahane cites Plato's Laws, 771c. 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(n). Asterisks indicate superior highly composite numbers.
Order HCNn prime factorization primeexponents numberof primefactors primorial factorization 1 1 0 1 2 2* 1 1 2 3 4 2 2 3 4 6* 1,1 2 4 5 12* 2,1 3 6 6 24 3,1 4 8 7 36 2,2 4 9 8 48 4,1 5 10 9 60* 2,1,1 4 12 10 120* 3,1,1 5 16 11 180 2,2,1 5 18 12 240 4,1,1 6 20 13 360* 3,2,1 6 24 14 7204,2,1 7 30 15 840 3,1,1,1 6 32 16 1260 2,2,1,1 6 36 17 1680 4,1,1,1 7 40 18 2520* 3,2,1,1 7 48 19 5040* 4,2,1,1 8 60 20 7560 3,3,1,1 8 64 21 10080 5,2,1,1 9 72 22 15120 4,3,1,1 9 80 23 20160 6,2,1,1 10 84 24 25200 4,2,2,1 9 90 25 27720 3,2,1,1,1 8 96 26 45360 4,4,1,1 10 100 27 50400 5,2,2,1 10 108 28 55440* 4,2,1,1,1 9 120 29 83160 3,3,1,1,1 9 128 30 110880 5,2,1,1,1 10 144 31 166320 4,3,1,1,1 10 160 32 221760 6,2,1,1,1 11 168 33 277200 4,2,2,1,1 10 180 34 332640 5,3,1,1,1 11 192 35 498960 4,4,1,1,1 11 200 36 554400 5,2,2,1,1 11 216 37 665280 6,3,1,1,1 12 224 38 720720* 4,2,1,1,1,1 10 240 39 1081080 3,3,1,1,1,1 10 256 40 1441440* 5,2,1,1,1,1 11 288 41 2162160 4,3,1,1,1,1 11 320
The divisors of the first 20 highly composite numbers are shown below.
n Divisors of n 1 1 1 2 2 1, 2 4 3 1, 2, 4 6 4 1, 2, 3, 6 12 6 1, 2, 3, 4, 6, 12 24 8 1, 2, 3, 4, 6, 8, 12, 24 36 9 1, 2, 3, 4, 6, 9, 12, 18, 36 48 10 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 60 12 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 120 16 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 180 18 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 240 20 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240 360 24 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 720 30 1, 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, 720 840 32 1, 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 1260 36 1, 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 1680 40 1, 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 2520 48 1, 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 5040 60 1, 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 7560 64 1, 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 highly composite number: 10080 10080 = (2 × 2 × 2 × 2 × 2) × (3 × 3) × 5 × 71 × 10080 2 × 5040 3 × 3360 4 × 2520 5 × 2016 6 × 1680 7 × 1440 8 × 1260 9 × 1120 10 × 1008 12 × 840 14 × 720 15 × 672 16 × 630 18 × 560 20 × 504 21 × 480 24 × 420 28 × 360 30 × 336 32 × 315 35 × 288 36 × 280 40 × 252 42 × 240 45 × 224 48 × 210 56 × 180 60 × 168 63 × 160 70 × 144 72 × 140 80 × 126 84 × 120 90 × 112 96 × 105Note: Numbers in bold are themselves highly composite numbers. Only the twentieth highly composite number 7560 (= 3 × 2520) is absent.10080 is a so-called 7-smooth number .
The 15,000-th highly composite number is the product of 230 primes:
where is the th successive prime number, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is ). 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 (2, 3, 5, ...); 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 (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four 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 (for instance 18 = 21 × 32 may be replaced with 12 = 22 × 31; both have six 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 (60) which has the same number of divisors.
Asymptotic growth and density If Q(x) 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
Sándor et al. (2006) p. 45
Related sequences
Highly composite numbers greater than 6 are also abundant numbers. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800.
10 of the first 38 highly composite numbers are superior highly composite numbers. The sequence of highly composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors .
Highly composite numbers whose number of divisors is also a highly composite number are 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 . It is known that this sequence is complete.
A positive integer n is a largely composite number if d(n) ≥ d(m) for all m ≤ n. The counting function QL(x) of largely composite numbers satisfies
for positive c and d with .Sándor et al. (2006) p. 46
Because the prime factorization of a highly composite number uses all of the first k primes, every highly composite number must be a practical number.. Due to their ease of use in calculations involving fractions, many of these numbers are used in traditional systems of measurement and engineering designs.
See also Superior highly composite number Highly totient number Table of divisors Euler's totient function Round number Smooth number
Notes
References
Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.
External links
Algorithm for computing Highly Composite Numbers First 10000 Highly Composite Numbers as factors Achim Flammenkamp, First 779674 HCN with sigma, tau, factors Online Highly Composite Numbers Calculator 5040 and other Anti-Prime Numbers - Dr. James Grime by Dr. James Grime for Numberphile