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Highly composite number

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Highly composite number

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept can be traced back to Plato, who set 5040 as the ideal number of citizens in a city because 5040 has more divisors than smaller numbers.[1]

The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer.

Examples

The initial or smallest 38 highly composite numbers are listed in the table below. (sequence A002182 in OEIS)

Order HCN
n
prime
factorization
prime
exponents
prime
factors
d(n) primorial
factorization
1 1 0 1
2* 2 2 1 1 2 2
3 4 2^2 2 2 3 2^2
4* 6 2\cdot 3 1,1 2 4 6
5* 12 2^2\cdot 3 2,1 3 6 2\cdot 6
6 24 2^3\cdot 3 3,1 4 8 2^2\cdot 6
7 36 2^2\cdot 3^2 2,2 4 9 6^2
8 48 2^4\cdot 3 4,1 5 10 2^3\cdot 6
9* 60 2^2\cdot 3\cdot 5 2,1,1 4 12 2\cdot 30
10* 120 2^3\cdot 3\cdot 5 3,1,1 5 16 2^2\cdot 30
11 180 2^2\cdot 3^2\cdot 5 2,2,1 5 18 6\cdot 30
12 240 2^4\cdot 3\cdot 5 4,1,1 6 20 2^3\cdot 30
13* 360 2^3\cdot 3^2\cdot 5 3,2,1 6 24 2\cdot 6\cdot 30
14 720 2^4\cdot 3^2\cdot 5 4,2,1 7 30 2^2\cdot 6\cdot 30
15 840 2^3\cdot 3\cdot 5\cdot 7 3,1,1,1 6 32 2^2\cdot 210
16 1260 2^2\cdot 3^2\cdot 5\cdot 7 2,2,1,1 6 36 6\cdot 210
17 1680 2^4\cdot 3\cdot 5\cdot 7 4,1,1,1 7 40 2^3\cdot 210
18* 2520 2^3\cdot 3^2\cdot 5\cdot 7 3,2,1,1 7 48 2\cdot 6\cdot 210
19* 5040 2^4\cdot 3^2\cdot 5\cdot 7 4,2,1,1 8 60 2^2\cdot 6\cdot 210
20 7560 2^3\cdot 3^3\cdot 5\cdot 7 3,3,1,1 8 64 6^2\cdot 210
21 10080 2^5\cdot 3^2\cdot 5\cdot 7 5,2,1,1 9 72 2^3\cdot 6\cdot 210
22 15120 2^4\cdot 3^3\cdot 5\cdot 7 4,3,1,1 9 80 2\cdot 6^2\cdot 210
23 20160 2^6\cdot 3^2\cdot 5\cdot 7 6,2,1,1 10 84 2^4\cdot 6\cdot 210
24 25200 2^4\cdot 3^2\cdot 5^2\cdot 7 4,2,2,1 9 90 2^2\cdot 30\cdot 210
25 27720 2^3\cdot 3^2\cdot 5\cdot 7\cdot 11 3,2,1,1,1 8 96 2\cdot 6\cdot 2310
26 45360 2^4\cdot 3^4\cdot 5\cdot 7 4,4,1,1 10 100 6^3\cdot 210
27 50400 2^5\cdot 3^2\cdot 5^2\cdot 7 5,2,2,1 10 108 2^3\cdot 30\cdot 210
28* 55440 2^4\cdot 3^2\cdot 5\cdot 7\cdot 11 4,2,1,1,1 9 120 2^2\cdot 6\cdot 2310
29 83160 2^3\cdot 3^3\cdot 5\cdot 7\cdot 11 3,3,1,1,1 9 128 6^2\cdot 2310
30 110880 2^5\cdot 3^2\cdot 5\cdot 7\cdot 11 5,2,1,1,1 10 144 2^3\cdot 6\cdot 2310
31 166320 2^4\cdot 3^3\cdot 5\cdot 7\cdot 11 4,3,1,1,1 10 160 2\cdot 6^2\cdot 2310
32 221760 2^6\cdot 3^2\cdot 5\cdot 7\cdot 11 6,2,1,1,1 11 168 2^4\cdot 6\cdot 2310
33 277200 2^4\cdot 3^2\cdot 5^2\cdot 7\cdot 11 4,2,2,1,1 10 180 2^2\cdot 30\cdot 2310
34 332640 2^5\cdot 3^3\cdot 5\cdot 7\cdot 11 5,3,1,1,1 11 192 2^2\cdot 6^2\cdot 2310
35 498960 2^4\cdot 3^4\cdot 5\cdot 7\cdot 11 4,4,1,1,1 11 200 6^3\cdot 2310
36 554400 2^5\cdot 3^2\cdot 5^2\cdot 7\cdot 11 5,2,2,1,1 11 216 2^3\cdot 30\cdot 2310
37 665280 2^6\cdot 3^3\cdot 5\cdot 7\cdot 11 6,3,1,1,1 12 224 2^3\cdot 6^2\cdot 2310
38* 720720 2^4\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13 4,2,1,1,1,1 10 240 2^2\cdot 6\cdot 30030

The table below shows all the divisors of one of these numbers.

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 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). [2]

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

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:

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.

Any factor of n must have the same or lesser multiplicity in each prime:

p_1^{d_1} \times p_2^{d_2} \times \cdots \times p_k^{d_k}, 0 \leq d_i \leq c_i, 0 < i \leq k

So the number of divisors of n is:

d(n) = (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. 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.

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

\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[3]

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 \ .

Related sequences

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.

10 of the first 38 highly composite numbers are superior highly composite numbers. 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).

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]

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.[6] 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.

See also

Notes

  1. ^  . Kahane cites Plato's Laws, 771c.
  2. ^ Flammenkamp, Achim, Highly Composite Numbers .
  3. ^ Sándor et al (2006) p.45
  4. ^ Sándor et al (2006) p.46
  5. ^  
  6. ^ Srinivasan, A. K. (1948), "Practical numbers" (PDF),  .

References

External links

  • Weisstein, Eric W., "Highly Composite Number", MathWorld.
  • Algorithm for computing Highly Composite Numbers
  • First 10000 Highly Composite Numbers
  • Achim Flammenkamp, First 779674 HCN with sigma,tau,factors
  • Online Highly Composite Numbers Calculator
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