12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Highly composite number | 1/2 | https://en.wikipedia.org/wiki/Highly_composite_number | reference | science, encyclopedia | 2026-05-05T08:15:00.548879+00:00 | kb-cron |
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N. For example, 6 is highly composite because d(6) = 4, and for n = 1,2,3,4,5, you get d(n) = 1,2,2,3,2, 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. 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 (sequence A002182 in the OEIS). The number of divisors is given in the column labeled d(n). Asterisks indicate superior highly composite numbers.
The divisors of the first 20 highly composite numbers are shown below.
The table below shows all 72 divisors of 10080 by writing it as a product of two numbers in 36 different ways.
The 15,000-th highly composite number 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
⋯
a
229
,
{\displaystyle 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
{\displaystyle a_{n}}
is the
n
{\displaystyle n}
th successive prime number, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is
2
14
×
3
9
×
5
6
×
⋯
×
1451
{\displaystyle 2^{14}\times 3^{9}\times 5^{6}\times \cdots \times 1451}
). More concisely, it is the product of seven distinct primorials:
b
0
5
b
1
3
b
2
2
b
4
b
7
b
18
b
229
,
{\displaystyle b_{0}^{5}b_{1}^{3}b_{2}^{2}b_{4}b_{7}b_{18}b_{229},}
where
b
n
{\displaystyle b_{n}}
is the primorial
a
0
a
1
⋯
a
n
{\displaystyle a_{0}a_{1}\cdots a_{n}}
.
== 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
×
p
2
c
2
×
⋯
×
p
k
c
k
{\displaystyle n=p_{1}^{c_{1}}\times p_{2}^{c_{2}}\times \cdots \times p_{k}^{c_{k}}}
where
p
1
<
p
2
<
⋯
<
p
k
{\displaystyle p_{1}<p_{2}<\cdots <p_{k}}
are prime, and the exponents
c
i
{\displaystyle c_{i}}
are positive integers. Any factor of n must have the same or lesser multiplicity in each prime:
p
1
d
1
×
p
2
d
2
×
⋯
×
p
k
d
k
,
0
≤
d
i
≤
c
i
,
0
<
i
≤
k
{\displaystyle 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
)
×
(
c
2
+
1
)
×
⋯
×
(
c
k
+
1
)
.
{\displaystyle d(n)=(c_{1}+1)\times (c_{2}+1)\times \cdots \times (c_{k}+1).}
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
c
1
≥
c
2
≥
⋯
≥
c
k
{\displaystyle 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 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
(
log
x
)
a
≤
Q
(
x
)
≤
(
log
x
)
b
.
{\displaystyle (\log x)^{a}\leq Q(x)\leq (\log 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
1.13682
<
lim inf
x
→
∞
log
Q
(
x
)
log
log
x
≤
1.44
{\displaystyle 1.13682<\liminf _{x\,\to \,\infty }{\frac {\log Q(x)}{\log \log x}}\leq 1.44\ }
and