Highly Composite Numbers

Today I turned 25200 days old and I was surprised to find that this number has a staggering 347 entries in the Online Encyclopaedia of Integer Sequences (OEIS). Most numbers of this size are lucky to have more than a dozen entries. So what's so special about 25200? Well, it turns out to be a highly composite number, a term first coined by Ramanujan in 1915 and defined as a number that sets a record for the highest number of factors (in this case 90). Here is a table from Wikipedia showing details for the first 38 highly composite numbers (sequence A002182 in the OEIS).

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

All highly composite numbers are products of primorials as can be see from rightmost column of the table. In the case of 25200, the primorial factorisation is \( 2^2 \times 30 \times 210 \). There is a formula for calculating the number of factors for a number n:$$ \text{If }n=\prod_{i=1}^k p_i \, c^i \text{ then } d(n)=\prod_{i=1}^k (c^i+1)$$For example: $$ 25200=2^4\cdot 3^2\cdot 5^2\cdot 7 $$ $$ d(25200)=(4+1) \cdot (2+1) \cdot (2+1) \cdot (1+1) = 5 \cdot 3 \cdot 3 \cdot 2 = 90 $$The sequence of indices is non-increasing when the prime factor bases are placed in ascending order (4, 2, 2, 1 in the case of 25200). The final index is always 1 except in the cases of 4 and 36 where it is 2, thus making 1, 2 and 4 the only square, highly composite numbers.