Highly composite number

This article is about numbers having many divisors. For numbers factorized only to powers of 2, 3, 5 and 7 (also named 7-smooth numbers), see Smooth number.

First four: 1, 2, 4, 6

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it.^[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 the OEIS). The number of divisors is given in the column labeled d(n).

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

A positive integer n is a largely composite number if d(n) ≥ d(m) for all m ≤ n. The counting function Q_L(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.

Notes

↑ Kahane, Jean-Pierre (February 2015), "Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'øeuvre", Notices of the American Mathematical Society, 62 (2): 136–140 . Kahane cites Plato's Laws, 771c.
↑ Flammenkamp, Achim, Highly Composite Numbers .
↑ Sándor et al (2006) p.45
↑ Sándor et al (2006) p.46
↑ Nicolas, Jean-Louis (1979). "Répartition des nombres largement composés". Acta Arith. (in French). 34: 379–390. Zbl 0368.10032.
↑ Srinivasan, A. K. (1948), "Practical numbers" (PDF), Current Science, 17: 179–180, MR 0027799 .

References

Ramanujan, S. (1915). "Highly composite numbers". Proc. London Math. Soc. (2). 14: 347–409. doi:10.1112/plms/s2_14.1.347. JFM 45.1248.01. (online)
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 45–46. ISBN 1-4020-4215-9. Zbl 1151.11300.
Erdös, P. (1944). "On highly composite numbers" (PDF). Journal of the London Mathematical Society. Second Series. 19: 130–133. doi:10.1112/jlms/19.75_part_3.130. MR 0013381.
Alaoglu, L.; Erdös, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. MR 0011087.
Ramanujan, Srinivasa (1997). "Highly composite numbers" (PDF). Ramanujan Journal. 1 (2): 119–153. doi:10.1023/A:1009764017495. MR 1606180. Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.

External links

Weisstein, Eric W. "Highly Composite Number". MathWorld.

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual

Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas

With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

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centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

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By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

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