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Highly composite number

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Demonstration, with Cuisenaire rods, of the first four highly composite numbers: 1, 2, 4, 6

A highly composite number is a positive integer that has more divisors than all smaller positive integers. 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.[1]

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.[2] Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040.[3]

Examples

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The first 41 highly composite numbers are listed in the table below (sequence A002182 in the OEIS). The number of divisors is given in the column labeled d(n). Asterisks indicate superior highly composite numbers.

Order HCN
n
prime
factorization
prime
exponents
number
of prime
factors
d(n) 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
14 720 4,2,1 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 19 highly composite numbers are shown below.

n d(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

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  ×  7
1
×
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
×
105
Note:  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 (sequence A002473 in the OEIS).

The 15,000th 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 (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 .[4]

Prime factorization

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Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are labelled in bold and superior highly composite numbers are starred. In the SVG file, hover over a bar to see its statistics.

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

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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

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Euler diagram of numbers under 100:
   Superabundant and highly composite
   Weird
   Perfect

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 (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS).

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 (sequence A189394 in the OEIS).

It is extremely likely that this sequence is complete.

A positive integer n is a largely composite number if d(n) ≥ d(m) for all mn. The counting function QL(x) of largely composite numbers satisfies

for positive c and d with .[6][7]

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.[8] 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

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Notes

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  1. ^ Ramanujan, S. (1915). "Highly composite numbers" (PDF). Proc. London Math. Soc. Series 2. 14: 347–409. doi:10.1112/plms/s2_14.1.347. JFM 45.1248.01.
  2. ^ Kahane, Jean-Pierre (February 2015), "Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'oeuvre", Notices of the American Mathematical Society, 62 (2): 136–140. Kahane cites Plato's Laws, 771c.
  3. ^ Vardoulakis, Antonis; Pugh, Clive (September 2008), "Plato's hidden theorem on the distribution of primes", The Mathematical Intelligencer, 30 (3): 61–63, doi:10.1007/BF02985381.
  4. ^ Flammenkamp, Achim, Highly Composite Numbers.
  5. ^ Sándor et al. (2006) p. 45
  6. ^ Sándor et al. (2006) p. 46
  7. ^ Nicolas, Jean-Louis (1979). "Répartition des nombres largement composés". Acta Arith. (in French). 34 (4): 379–390. doi:10.4064/aa-34-4-379-390. Zbl 0368.10032.
  8. ^ Srinivasan, A. K. (1948), "Practical numbers" (PDF), Current Science, 17: 179–180, MR 0027799.

References

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