An Approach to Counting Odd Number

by Frank C. Fung - 1st published in October, 2008.

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Section IV - Counting { Odd Numbers } up to [ 19,863 ]

Summary for the Section :

• In this Section , we shall provide a worked-example in an attempt to gain some insights .

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Counting { odd numbers } up to [ 19,683 ] :

• We note , first-of all , that [ 19,683 ] is simply [ '3' raised to the 9th power ] , i.e. :
  • [ 19,683 ] = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 3^9 .

  Secondly , [ Category Zero ] does have , and will always have , only one (1) single member , namely :

  • the { odd number } [ 1 ] , which is not considered a { prime number } .

    And as such , [ Category Zero ] is always treated as [ contant ] and shall be omitted from our { summary-count } table below .

• Let us now break-down the { odd numbers } in the range from [ 3 ] to [ 19,683 ] ,
  • into the { categories of odd numbers } as previously defined , via this table below :

    Category of Odd Number	Number of members in category	* * *	Smallest member in category	Largest member in category	Largest member in category factorized
    Category I	2,226		3	19,681	19,681 is prime
    Category II	3,798		9	19,679	11 x 1,789
    Category III	2,451		27	19,677	3 x 7 x 937
    Category IV	970		81	19,647	3 x 3 x 37 x 59
    Category V	296		243	19,665	3 x 3 x 5 x 19 x 23
    Category VI	78		729	19,575	3 x 3 x 3 x 5 x 5 x 29
    Category VII	17		2,187	18,711	3 x 3 x 3 x 3 x 3 x 7 x 11
    Category VIII	4		6,561	18,225	3 x 3 x 3 x 3 x 3 x 3 x 5 x 5
    Category IX	1		19,683	19,683	3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3
    TOTAL	9,841

    ( The data above is generated via a Visual Basic computer program ) .

  And we note here that :

  • [ 19,683 ] = 2 x [ 9,841 ] + 1

    so that our { Total Count of [ 9,841 ] } here does cover the full range of { odd numbers } from [ 3 ] to [ 19,683 ] .

• One quick observation here :
  • The { smallest member } of each individual { category } is always a { pure power of [ 3 ] } .

    This should be quite obvious , arising from the fact that [ 3 ] is the smallest { odd-prime } ,

    • and no further explanation is deemed necessary here .

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• In the next 3 Sections , we shall be looking into specific { counting procedures } for :
  • { Category II Odd-Numbers } ,
  • { Category III Odd-Numbers } , and
  • { Category VII / VIII / IX Odd-Numbers } .

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go to the next section : Section V - Details on { Category II Odd Numbers }

go to the last section : Section III - Defining the { Counting Categories }

return to the { Counting Odd-Numbers } HomePage

original dated 2008-10-04