35. HIGHLY COMPOSITE NUMBERS | |||
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We know

A prime has only two divisors, itself and 1;

A composite has more than two.

if

we can write as

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 | ||

Ramanujan defines,

From the above list,

are

Ramanujan lists the first 102 of these,

2, 4, 6, 12, 24, 36, 48, 60, 120, ..., 674632838800.

N = **674632838800** = 2^{6}.3^{4}.5^{2}.7^{2}.11.13.17.19.23

d(674632838800) = 10080 = 2^{5} . 3^{2} . 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,**...**.

d(674632838800) = 10080 = 2

A bonus result is dd(

Then Ramanujan goes on to

The first few of these superior highly composite numbers: 2, 6, 12, 60, 120, 360, 2520, 5040, 554400, 720720,