Highly composite number

A highly composite number in math (also called anti-prime) is a real number with more divisors than any smaller real number smaller than it.

Jean-Pierre Kahane thought that Plato might have known about highly composite numbers. This is because he chose 5040 as a good number of citizens in a city as 5040 has more divisor than any numbers less than it.

Examples

The first 38 highly composite numbers are listed in the table below (sequence A002182 in OEIS). The number of divisors is given in the column labeled d(n). The letters with asterisks are also superior highly composite numbers.

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

The divisor of the first 15 highly composite numbers are shown below.

n	d(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

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

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:  The numbers in bold are also highly composite numbers. 10080 is often referred to as a 7-smooth number (sequence A002473 in OEIS).

Similar sequences

Every highly composite number that is bigger than 6 is also an abundant number. Not all highly composite numbers are also Harshad numbers, however most of them are the same. The first highly composite number that is not a Harshad number is 245,044,800. This number's digit's sum is 27. 27, however, doesn't divide into 245,044,800 evenly.

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

Highly Composite Number Media

• 

  Demonstration, with Cuisenaire rods, of the first four: 1, 2, 4, 6

• 

  Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are labelled in bold and superior highly composite numbers are starred. In the SVG file, hover over a bar to see its statistics.

Related pages

• Abundant number
• Euler's totient function
• Harshad number

References

• 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 (75_Part_3): 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. JSTOR 1990319. MR 0011087.
• Ramanujan, Srinivasa (1997). "Highly composite numbers" (PDF). Ramanujan Journal. 1 (2): 119–153. doi:10.1023/A:1009764017495. MR 1606180. S2CID 115619659. Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.

Other websites

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

• 5040 and other Anti-Prime Numbers - Dr. James Grime by Dr. James Grime for Numberphile