Superparticular Craziness

For my classical arithmetic and geometry class I'm studying Nicomachus's Intro to arithmetic.  It's pretty funky stuff.  Today I studied superparticular numbers - really mind bending.  Not sure of their significance but cool!

Check this out

If 

x = the superparticular number

n = the number being compared to x, or the subsuperparticular number

y = the fraction of n that is a factor of x 

Then 

x= n + yn

For example when y = 1/2 the equation would be 

x = n + ½ n or 6 = 4 + ½ (4).

If y is 1/2 the equation will work with the following pairs of numbers

3:2

6:4

9:6

12:8

15:10

If y is 1/3 the equation works with these numbers

4:3

8:6

12:9

16:12

20:15

If y is 1/4 it works with these

5:4

10:8

15:12

20:16

25:20

If w is 1/5 it works with

6:5

12:10

18:15

24:20

30:25

Look at these number pairs arranged this way

(n is 1/2)            3:2         6:4         9:6        12:8        15:10

(n is 1/3)            4:3         8:6         12:9      16:12      20:15

(n is 1/4)            5:4        10:8        15:12    20:16      25:20

(n is 1/5)            6:5        12:10      18:15    24:20      30:25

For the # below        +1          +2            +3         +4           +5