35. HIGHLY COMPOSITE NUMBERS
 
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We know Primes and Composites.
A prime has only two divisors, itself and 1;
A composite has more than two.

In other words:
if p is a prime; c is a composite; and
d(x) is Number of divisors of x,
we can write as d(p) = 2 and d(c) > 2.

Look at the number of divisors of numbers from 2 upwards.

Number Divisors No. of Divisors
N Divisors d(N)

2 1, 2 2
3 1, 3 2
4 1, 2, 4 2
5 1, 5 2
6 1, 2, 3, 6 2
7 1, 7 2
8 1, 2, 4, 8 2
9 1, 3, 9 2
10 1, 2, 5 10 2
11 1, 11 2
12 1, 2, 3, 4, 6, 12 2
13 1, 13 2


A Highly Composite Number
Ramanujan defines, A highly composite number as a number whose number of divisors exceeds that of all its predecessors.

From the above list, 2, 4, 6, 12
are highly composite numbers.

Ramanujan lists the first 102 of these,
2, 4, 6, 12, 24, 36, 48, 60, 120, ..., 674632838800.

N = 674632838800 = 26.34.52.72.11.13.17.19.23
d(674632838800) = 10080 = 25 . 32 . 5 . 7.

A bonus result is dd(N) = d(d(N)) = 72

Then Ramanujan goes on to
Superior Highly Composite Numbers.
The first few of these superior highly composite numbers: 2, 6, 12, 60, 120, 360, 2520, 5040, 554400, 720720, ....

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