Highly composite number

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A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. The term was coined by Ramanujan (1915), who showed that there are infinitely many such numbers. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer.

The initial or smallest twenty-six highly composite numbers are listed in the table at right.

Order n prime number
factorization
number of
divisors of n
1 1 1
2 2 2 2
3 4 2^2 3
4 6 2\cdot 3 4
5 12 2^2\cdot 3 6
6 24 2^3\cdot 3 8
7 36 2^2\cdot 3^2 9
8 48 2^4\cdot 3 10
9 60 2^2\cdot 3\cdot 5 12
10 120 2^3\cdot 3\cdot 5 16
11 180 2^2\cdot 3^2\cdot 5 18
12 240 2^4\cdot 3\cdot 5 20
13 360 2^3\cdot 3^2\cdot 5 24
14 720 2^4\cdot 3^2\cdot 5 30
15 840 2^3\cdot 3\cdot 5\cdot 7 32
16 1,260 2^2\cdot 3^2\cdot 5\cdot 7 36
17 1,680 2^4\cdot 3\cdot 5\cdot 7 40
18 2,520 2^3\cdot 3^2\cdot 5\cdot 7 48
19 5,040 2^4\cdot 3^2\cdot 5\cdot 7 60
20 7,560 2^3\cdot 3^3\cdot 5\cdot 7 64
21 10,080 2^5\cdot 3^2\cdot 5\cdot 7 72
22 15,120 2^4\cdot 3^3\cdot 5\cdot 7 80
23 20,160 2^6\cdot 3^2\cdot 5\cdot 7 84
24 25,200 2^4\cdot 3^2\cdot 5^2\cdot 7 90
25 27,720 2^3\cdot 3^2\cdot 5\cdot 7\cdot 11 96
26 45,360 2^4\cdot 3^4\cdot 5\cdot 7 100

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


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. If we decompose a number n in prime factors like this:

n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}\qquad (1)

where p_1 < p_2 < \cdots < p_k are prime, and the exponents c_i are positive integers, then the number of divisors of n is exactly

(c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1).\qquad (2)

Hence, for n to be a highly composite number,

  • 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 c_1 \geq c_2 \geq \cdots \geq c_k; 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. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. 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.[1]

Highly composite numbers higher than 6 are also abundant numbers. One need only look at the three or four highest 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 HCN that is not a Harshad number is 245,044,800, which has a digit sum of 27, but 27 does not divide evenly into 245,044,800.

Many of these numbers are used in traditional systems of measurement, and tend to be used in engineering designs, due to their ease of use in calculations involving fractions.

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

\ln(x)^a \le Q(x) \le \ln(x)^b \, .

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[2]

1.13862 < \liminf \frac{\log Q(x)}{\log\log x} \le 1.44 \

and

\limsup \frac{\log Q(x)}{\log\log x} \le 1.71 \ .

Examples[edit]

The highly composite number: 10,080
10,080 = (2 × 2 × 2 × 2 × 2)  ×  (3 × 3) ×  5  ×  7
By (2) above, 10,080 has exactly seventy-two divisors.
1
×
10,080
2
×
5,040
3
×
3,360
4
×
2,520
5
×
2,016
6
×
1,680
7
×
1,440
8
×
1,260
9
×
1,120
10
×
1,008
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 OEIS).

The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:

a_0^{14} a_1^9 a_2^6 a_3^4 a_4^4 a_5^3 a_6^3 a_7^3 a_8^2 a_9^2 a_{10}^2 a_{11}^2 a_{12}^2 a_{13}^2 a_{14}^2 a_{15}^2 a_{16}^2 a_{17}^2 a_{18}^{2} a_{19} a_{20} a_{21}\cdots a_{229},

where a_n is the sequence of successive prime numbers, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is 2^{14}  \times 3^{9}  \times 5^6  \times \cdots \times 1451). [3]

Plot of the number of divisors of integers from 1 to 1000. The first 15 highly-composite numbers are in bold.

Prime factor subsets[edit]

For any highly composite number, if one takes any subset of prime factors for that number and their exponents, the resulting number will have more divisors than any smaller number that uses the same prime factors. For example for the highly composite number 720 which is 24 × 32 × 5 we can be sure that

  • 144 which is 24 × 32 has more divisors than any smaller number that has only the prime factors 2 and 3
  • 80 which is 24 × 5 has more divisors than any smaller number that has only the prime factors 2 and 5
  • 45 which is 32 × 5 has more divisors than any smaller number that has only the prime factors 3 and 5

If this were untrue for any particular highly composite number and subset of prime factors, we could exchange that subset of prime factors and exponents for the smaller number using the same prime factors and get a smaller number with at least as many divisors.

This property is useful for finding highly composite numbers.

Largely composite numbers[edit]

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

(\log x)^c \le \log Q_L(x) \le )\log x)^d \

for positive c,d with 0.2 \le c \le d \le 0.5.[4][5]

See also[edit]

References[edit]

  1. ^ Srinivasan, A. K. (1948), "Practical numbers", Current Science 17: 179–180, MR 0027799 .
  2. ^ Sándor et al (2006) p.45
  3. ^ Flammenkamp, Achim, Highly Composite Numbers .
  4. ^ Sándor et al (2006) p.46
  5. ^ Nicolas, Jean-Louis (1979). "Répartition des nombres largement composés". Acta Arith. (in French) 34: 379–390. Zbl 0368.10032. 

External links[edit]