Highly composite number

{{short description|Positive integer with more divisors than all smaller positive integers}}

{{about|numbers having many divisors|numbers factorized only to powers of 2, 3, 5 and 7 (also named 7-smooth numbers)|Smooth number}}

File:Highly composite number Cuisenaire rods 6.png, 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. 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 d(n)=1,2,2,3,2 for n=1,2,3,4,5 respectively.

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.{{cite journal | last1=Ramanujan | first1=S. | author1-link=Srinivasa Ramanujan | title=Highly composite numbers | jfm=45.1248.01 | doi=10.1112/plms/s2_14.1.347 | journal=Proc. London Math. Soc. |series=Series 2 | volume=14 | pages=347–409 | year=1915| url=http://ramanujan.sirinudi.org/Volumes/published/ram15.pdf}}

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.{{citation|first=Jean-Pierre|last=Kahane|author-link=Jean-Pierre Kahane|title=Bernoulli convolutions and self-similar measures after Erdős: A personal hors d'oeuvre|journal=Notices of the American Mathematical Society|date=February 2015|volume=62|issue=2|pages=136–140}}. Kahane cites Plato's Laws, 771c. Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040.{{citation|last1=Vardoulakis|first1=Antonis|last2=Pugh|first2=Clive|title=Plato's hidden theorem on the distribution of primes|journal=The Mathematical Intelligencer|date=September 2008|volume=30|issue=3|pages=61–63|doi=10.1007/BF02985381 |url=https://link.springer.com/article/10.1007/BF02985381}}.

Examples

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

class="wikitable" style="text-align:left"
Order ! HCN n ! prime factorization ! prime exponents ! number of prime factors ! {{abbr\|d(n)\|number of divisors of 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 \| $2^2\cdot 6\cdot 30030$
39 \| 1081080 \| $2^3\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13$ \|3,3,1,1,1,1 \| 10 \| 256 \| $6^2\cdot 30030$
40 \| 1441440* \| $2^5\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13$ \| 5,2,1,1,1,1 \| 11 \| 288 \| $2^3\cdot 6\cdot 30030$
41 \| 2162160 \| $2^4\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13$ \| 4,3,1,1,1,1 \| 11 \| 320 \| $2\cdot 6^2\cdot 30030$

class="wikitable" style="text-align:left"

Order

! HCN
n

! prime
factorization

! prime
exponents

! number
of prime
factors

! {{abbr|d(n)|number of divisors of n}}

! primorial
factorization

| 1

| 0

| 1

| 2*

| $2$

| 1

| 2

| $2$

| 4

| $2^2$

| 2

| 3

| $2^2$

| 6*

| $2\cdot 3$

|1,1

| 2

| 4

| $6$

| 12*

| $2^2\cdot 3$

|2,1

| 3

| 6

| $2\cdot 6$

| 24

| $2^3\cdot 3$

|3,1

| 4

| 8

| $2^2\cdot 6$

| 36

| $2^2\cdot 3^2$

|2,2

| 4

| 9

| $6^2$

| 48

| $2^4\cdot 3$

|4,1

| 5

| 10

| $2^3\cdot 6$

| 60*

| $2^2\cdot 3\cdot 5$

|2,1,1

| 4

| 12

| $2\cdot 30$

| 120*

| $2^3\cdot 3\cdot 5$

|3,1,1

| 5

| 16

| $2^2\cdot 30$

| 180

| $2^2\cdot 3^2\cdot 5$

|2,2,1

| 5

| 18

| $6\cdot 30$

| 240

| $2^4\cdot 3\cdot 5$

|4,1,1

| 6

| 20

| $2^3\cdot 30$

| 360*

| $2^3\cdot 3^2\cdot 5$

|3,2,1

| 6

| 24

| $2\cdot 6\cdot 30$

| 720

| $2^4\cdot 3^2\cdot 5$

|4,2,1

| 7

| 30

| $2^2\cdot 6\cdot 30$

| 840

| $2^3\cdot 3\cdot 5\cdot 7$

|3,1,1,1

| 6

| 32

| $2^2\cdot 210$

| 1260

| $2^2\cdot 3^2\cdot 5\cdot 7$

|2,2,1,1

| 6

| 36

| $6\cdot 210$

| 1680

| $2^4\cdot 3\cdot 5\cdot 7$

|4,1,1,1

| 7

| 40

| $2^3\cdot 210$

| 2520*

| $2^3\cdot 3^2\cdot 5\cdot 7$

|3,2,1,1

| 7

| 48

| $2\cdot 6\cdot 210$

| 5040*

| $2^4\cdot 3^2\cdot 5\cdot 7$

|4,2,1,1

| 8

| 60

| $2^2\cdot 6\cdot 210$

| 7560

| $2^3\cdot 3^3\cdot 5\cdot 7$

|3,3,1,1

| 8

| 64

| $6^2\cdot 210$

| 10080

| $2^5\cdot 3^2\cdot 5\cdot 7$

|5,2,1,1

| 9

| 72

| $2^3\cdot 6\cdot 210$

| 15120

| $2^4\cdot 3^3\cdot 5\cdot 7$

|4,3,1,1

| 9

| 80

| $2\cdot 6^2\cdot 210$

| 20160

| $2^6\cdot 3^2\cdot 5\cdot 7$

|6,2,1,1

| 10

| 84

| $2^4\cdot 6\cdot 210$

| 25200

| $2^4\cdot 3^2\cdot 5^2\cdot 7$

|4,2,2,1

| 9

| 90

| $2^2\cdot 30\cdot 210$

| 27720

| $2^3\cdot 3^2\cdot 5\cdot 7\cdot 11$

|3,2,1,1,1

| 8

| 96

| $2\cdot 6\cdot 2310$

| 45360

| $2^4\cdot 3^4\cdot 5\cdot 7$

|4,4,1,1

| 10

| 100

| $6^3\cdot 210$

| 50400

| $2^5\cdot 3^2\cdot 5^2\cdot 7$

|5,2,2,1

| 10

| 108

| $2^3\cdot 30\cdot 210$

| 55440*

| $2^4\cdot 3^2\cdot 5\cdot 7\cdot 11$

|4,2,1,1,1

| 9

| 120

| $2^2\cdot 6\cdot 2310$

| 83160

| $2^3\cdot 3^3\cdot 5\cdot 7\cdot 11$

|3,3,1,1,1

| 9

| 128

| $6^2\cdot 2310$

| 110880

| $2^5\cdot 3^2\cdot 5\cdot 7\cdot 11$

|5,2,1,1,1

| 10

| 144

| $2^3\cdot 6\cdot 2310$

| 166320

| $2^4\cdot 3^3\cdot 5\cdot 7\cdot 11$

|4,3,1,1,1

| 10

| 160

| $2\cdot 6^2\cdot 2310$

| 221760

| $2^6\cdot 3^2\cdot 5\cdot 7\cdot 11$

|6,2,1,1,1

| 11

| 168

| $2^4\cdot 6\cdot 2310$

| 277200

| $2^4\cdot 3^2\cdot 5^2\cdot 7\cdot 11$

|4,2,2,1,1

| 10

| 180

| $2^2\cdot 30\cdot 2310$

| 332640

| $2^5\cdot 3^3\cdot 5\cdot 7\cdot 11$

|5,3,1,1,1

| 11

| 192

| $2^2\cdot 6^2\cdot 2310$

| 498960

| $2^4\cdot 3^4\cdot 5\cdot 7\cdot 11$

|4,4,1,1,1

| 11

| 200

| $6^3\cdot 2310$

| 554400

| $2^5\cdot 3^2\cdot 5^2\cdot 7\cdot 11$

|5,2,2,1,1

| 11

| 216

| $2^3\cdot 30\cdot 2310$

| 665280

| $2^6\cdot 3^3\cdot 5\cdot 7\cdot 11$

|6,3,1,1,1

| 12

| 224

| $2^3\cdot 6^2\cdot 2310$

| 720720*

| $2^4\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13$

|4,2,1,1,1,1

| 10

| 240

| $2^2\cdot 6\cdot 30030$

| 1081080

| $2^3\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13$

|3,3,1,1,1,1

| 10

| 256

| $6^2\cdot 30030$

| 1441440*

| $2^5\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13$

| 5,2,1,1,1,1

| 11

| 288

| $2^3\cdot 6\cdot 30030$

| 2162160

| $2^4\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13$

| 4,3,1,1,1,1

| 11

| 320

| $2\cdot 6^2\cdot 30030$

The divisors of the first 20 highly composite numbers are shown below.

class="wikitable" ! n !! {{abbr\|d(n)\|number of divisors of 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
7560	64	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 54, 56, 60, 63, 70, 72, 84, 90, 105, 108, 120, 126, 135, 140, 168, 180, 189, 210, 216, 252, 270, 280, 315, 360, 378, 420, 504, 540, 630, 756, 840, 945, 1080, 1260, 1512, 1890, 2520, 3780, 7560

The table below shows all 72 divisors of 10080 by writing it as a product of two numbers in 36 different ways.

class="wikitable" style="text-align:center;table-layout:fixed;"

colspan="6"| The highly composite number: 10080
10080 = (2 × 2 × 2 × 2 × 2) × (3 × 3) × 5 × 7

style="color:#000000;background:#ffffff;"

|style="line-height:1.4" height=64|1
×
10080

|style="line-height:1.4"| 2
×
5040

|style="line-height:1.4"| 3
×
3360

|style="line-height:1.4"| 4
×
2520

|style="line-height:1.4"| 5
×
2016

|style="line-height:1.4"| 6
×
1680

style="color:#000000;background:#ffffff;"

|style="line-height:1.4" height=64|7
×
1440

|style="line-height:1.4"| 8
×
1260

|style="line-height:1.4"| 9
×
1120

|style="line-height:1.4"| 10
×
1008

|style="line-height:1.4"| 12
×
840

|style="line-height:1.4"| 14
×
720

style="color:#000000;background:#ffffff;"

|style="line-height:1.4" height=64|15
×
672

|style="line-height:1.4"| 16
×
630

|style="line-height:1.4"| 18
×
560

|style="line-height:1.4"| 20
×
504

|style="line-height:1.4"| 21
×
480

|style="line-height:1.4"| 24
×
420

style="color:#000000;background:#ffffff;"

|style="line-height:1.4" height=64|28
×
360

|style="line-height:1.4"| 30
×
336

|style="line-height:1.4"| 32
×
315

|style="line-height:1.4"| 35
×
288

|style="line-height:1.4"| 36
×
280

|style="line-height:1.4"| 40
×
252

style="color:#000000;background:#ffffff;"

|style="line-height:1.4" height=64|42
×
240

|style="line-height:1.4"| 45
×
224

|style="line-height:1.4"| 48
×
210

|style="line-height:1.4"| 56
×
180

|style="line-height:1.4"| 60
×
168

|style="line-height:1.4"| 63
×
160

style="color:#000000;background:#ffffff;"

|style="line-height:1.4" height=64|70
×
144

|style="line-height:1.4"| 72
×
140

|style="line-height:1.4"| 80
×
126

|style="line-height:1.4"| 84
×
120

|style="line-height:1.4"| 90
×
112

|style="line-height:1.4"| 96
×
105

colspan="6"|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 {{OEIS|id=A002473}}.

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 $n$ th successive prime number, and all omitted terms (a₂₂ to a₂₂₈) are factors with exponent equal to one (i.e. the number is $2^{14} \times 3^{9} \times 5^6 \times \cdots \times 1451$ ). More concisely, it is the product of seven distinct primorials:

: $b_0^5 b_1^3 b_2^2 b_4 b_7 b_{18} b_{229},$

where $b_n$ is the primorial $a_0a_1\cdots a_n$ .{{citation

| last = Flammenkamp | first = Achim

| title = Highly Composite Numbers

| url = http://wwwhomes.uni-bielefeld.de/achim/highly.html}}.

Prime factorization

File:Highly_composite_numbers.svg

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:

: $n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}$

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:

: $d(n) = (c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1).$

Hence, for a highly composite number n,

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 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

: $(\log x)^a \le Q(x) \le (\log x)^b \, .$

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

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

and

: $\limsup \frac{\log Q(x)}{\log\log x} \le 1.71 \ .$ Sándor et al. (2006) p. 45

Related sequences

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 {{OEIS|id=A002182}} is a subset of the sequence of smallest numbers k with exactly n divisors {{OEIS|id=A005179}}.

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 {{OEIS|id=A189394}}.

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 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 and d with $0.2 \le c \le d \le 0.5$ .Sándor et al. (2006) p. 46{{cite journal | last=Nicolas | first=Jean-Louis | author-link=Jean-Louis Nicolas | title=Répartition des nombres largement composés | language=fr | zbl=0368.10032 | journal=Acta Arith. | volume=34 | issue=4 | pages=379–390 | year=1979 | doi=10.4064/aa-34-4-379-390 | doi-access=free}}

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.{{citation

| last = Srinivasan | first = A. K.

| title = Practical numbers

| journal = Current Science

| volume = 17

| year = 1948

| pages = 179–180

| mr=0027799

| url = http://www.ias.ac.in/jarch/currsci/17/179.pdf}}. Due to their ease of use in calculations involving fractions, many of these numbers are used in traditional systems of measurement and engineering designs.

Notes

References

{{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=Springer-Verlag | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | pages=45–46}}
{{cite journal

| last = Erdös | first = P. | author-link = Paul Erdős

| journal = Journal of the London Mathematical Society

| mr = 0013381

| pages = 130–133

| series = Second Series

| title = On highly composite numbers

| url = https://www.renyi.hu/~p_erdos/1944-04.pdf

| volume = 19

| issue = 75_Part_3 | year = 1944

| doi=10.1112/jlms/19.75_part_3.130}}

{{cite journal

| last1 = Alaoglu | first1 = L. | author1-link = Leonidas Alaoglu

| last2 = Erdös | first2 = P. | author2-link = Paul Erdős

| issue = 3

| journal = Transactions of the American Mathematical Society

| mr = 0011087

| pages = 448–469

| title = On highly composite and similar numbers

| url = https://www.renyi.hu/~p_erdos/1944-03.pdf

| volume = 56

| year = 1944 | doi=10.2307/1990319| jstor = 1990319 }}

{{cite journal

| last = Ramanujan | first = Srinivasa | author-link = Srinivasa Ramanujan

| doi = 10.1023/A:1009764017495

| issue = 2

| journal = Ramanujan Journal

| mr = 1606180

| pages = 119–153

| title = Highly composite numbers

| url = http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf

| volume = 1

| year = 1997| s2cid = 115619659 }} Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.

External links

{{MathWorld |urlname=HighlyCompositeNumber |title=Highly Composite Number}}
[https://web.archive.org/web/19980707133810/http://www.math.princeton.edu/~kkedlaya/math/hcn-algorithm.tex Algorithm for computing Highly Composite Numbers]
[https://web.archive.org/web/19980707133953/http://www.math.princeton.edu/~kkedlaya/math/hcn10000.txt.gz First 10000 Highly Composite Numbers as factors]
[http://wwwhomes.uni-bielefeld.de/achim/highly.html Achim Flammenkamp, First 779674 HCN with sigma, tau, factors]
[http://www.javascripter.net/math/calculators/highlycompositenumbers.htm Online Highly Composite Numbers Calculator]
[https://www.youtube.com/watch?v=2JM2oImb9Qg 5040 and other Anti-Prime Numbers - Dr. James Grime] by Dr. James Grime for Numberphile

Category:Integer sequences

Category:Mathematics-related lists

Highly composite number

Examples

Prime factorization

Asymptotic growth and density

Related sequences

See also

Notes

References

External links