S3. Defining the Counting Categories

An Approach to Counting Odd Numbers

Section III - Defining the { Counting Categories }

In this Section , we shall formally define the { counting categories } for our counting purposes .

For our counting exercises to follow , we shall be segregating the { odd numbers } into different { counting sub-groups } ,

and these { counting sub-groups } are categorized as follows :

Category Zero	contains the single-and-unique { Odd Number } [ 1 ] , which is not considered a { prime number } .
Category I	{ Odd Numbers } that are the product of one (1) single { odd prime number } .
Category II	{ Odd Numbers } that are the product of two (2) { odd prime numbers } .
Category III	{ Odd Numbers } that are the product of three (3) { odd prime numbers } .
Category IV	{ Odd Numbers } that are the product of four (4) { odd prime numbers } .
And so-on-and-so-forth

A few quick comments here , for better clarity :

{ Category I Odd-Numbers } are always { prime numbers } , i.e. :
- [ 3 ] , [ 5 ] , [ 7 ] , [ 11 ] , [ 13 ] , [ 17 ] , [ 19 ] , [ 23 ] , etc.
Examples of { Category II Odd-Numbers } are as follows :
- [ 9 ] , [ 15 ] , [ 21 ] , [ 25 ] , [ 33 ] , [ 35 ] , [ 39 ] , [ 49 ] , etc. ,
  
  with :
  
  [ 9 = 3 x 3 ] , [ 15 = 3 x 5 ] , [ 21 = 3 x 7 ] , [ 25 = 5 x 5 ] , etc.
Examples of { Category III Odd-Numbers } are as follows :
- [ 27 ] , [ 45 ] , [ 63 ] , [ 75 ] , [ 105 ] , [ 125 ] , [ 147 ] , [ 175 ] , etc.
  
  with :
  
  [ 27 = 3 x 3 x 3 ] , [ 45 = 3 x 3 x 5 ] , [ 63 = 3 x 3 x 7 ] , [ 75 = 3 x 5 x 5 ] , etc.

In the next Section , we shall go-thru the execise of counting { odd numbers } up to [ 19,683 ] :
- and the purpose of these { counting categories } will become more apparent .