An Approach to Counting Odd Numbers

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

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Section VII - Details on { Category VII / VIII / IX Odd-Numbers }

Summary for the Section :

• In this Section , we shall provide further details on { Category VII / VIII / IX Odd-Numbers } .

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Statistics for { Category VII / VIII / IX Odd-Numbers } up to [ 19,683 ] :

• Let us first bring-back the statistics on these 3 categories , from Section IV :

  Category of Odd Number	Number of members in category	* * *	Smallest member in category	Largest member in category	Largest member in category factorized
  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

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{ Category IX Odd-Numbers } :

• It should be obvious , and redundant , that :
  • the only { Category IX Odd- Numbers } smaller than or equal to [ 19,683 ] is indeed [ 19,683 ] .

  The [ count ] is [ 1 ] , and no further comments here .

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{ Category VIII Odd-Numbers } :

• We note , first-of-all , that there are only 4 { Category II Odd-Numbers } less than [ 27 ] , namely :
  • [ 9 ] = 3 x 3 ,
  • [ 15 ] = 3 x 5 ,
  • [ 21 ] = 3 x 7 , and
  • [ 25 ] = 5 x 5 .

  Secondly , [ 729 ] is equal to [ '3' rasied to the 6th power ] .

• Multiplying the four (4) [ numbers ] above by [ 729 ] will then yield the four (4) [ desired numbers ] for { Category VIII } .

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{ Category VII Odd-Numbers } :

• This is slightly more complicated and we shall go thru 2 different { analysis procedures } here to illustrate a special feature :
  • { Analysis Procedure #1 } :

    We set-up , first-of-all , 4 sets of numbers :

    • { Set A } - containing 14 { Category II Odd-Numbers } that are less than [ 81 ] :
      • [ 9 ] = 3 x 3 ,
      • [ 15 ] = 3 x 5 ,
      • [ 21 ] = 3 x 7 ,
      • [ 25 ] = 5 x 5 ,
      • [ 33 ] = 3 x 11 ,

      • [ 35 ] = 5 x 7 ,
      • [ 39 ] = 3 x 13 ,
      • [ 49 ] = 7 x 7 ,
      • [ 51 ] = 3 x 17 ,
      • [ 55 ] = 5 x 11 ,

      • [ 57 ] = 3 x 19 ,
      • [ 65 ] = 5 x 13 ,
      • [ 69 ] = 3 x 23 , and
      • [ 77 ] = 7 x 11 .

    • { Set B } - containing 2 selected { Category III Odd-Numbers } that are less than [ 243 ] :
      • [ 125 ] = 5 x 5 x 5 , and
      • [ 175 ] = 5 x 5 x 7 .

    • { Set C } - containing 1 selected { Category IV Odd-Number } that is less than [ 729 ] :
      • [ 625 ] = 5 x 5 x 5 x 5 .

    The selection criteria for { Set B } and { Set C } are then as follows :

    • Each of the 2 [ numbers ] in { Set B } cannot be equivalent to :
      • [ 3 ] x [ any of the 14 numbers in { Set A } ] .

    • The one [ number ] in { Set C } cannot be equivalent to :
      • [ 3 ] x [ any of the 2 numbers in { Set B } ] , or
      • a direct multiple of { [ 3 ] x [ any of the 14 numbers in { Set A } ] } .

    And we note at this point that :

    • [ 243 ] is equal to [ '3' raised to the 5th power ] ,
    • [ 81 ] is equal to [ '3' raised to the 4th power ] ,
    • [ 27 ] is equal to [ '3' raised to the 3rd power ] ;

      so that we can arrive at the 17 { Category VII Odd-Numbers } via the following :

      • multiplying the [ 14 Category-II numbers in { Set A } ] by [ 243 ] ,
      • multiplying the [ 2 Category-III numbers in { Set B } ] by [ 81 ] ,
      • multiplying the [ 1 Category-IV number in { Set C } ] by [ 27 ] .

  • { Analysis Procedure #2 } :

    We set-up , at this point , { Set D } containing 17 { Category IV Odd-Numbers } less than [ 729 ] :

    • [ 81 ] = 3 x 3 x 3 x 3 ,
    • [ 135 ] = 3 x 3 x 3 x 5 ,
    • [ 189 ] = 3 x 3 x 3 x 7 ,
    • [ 225 ] = 3 x 3 x 5 x 5 ,
    • [ 297 ] = 3 x 3 x 3 x 11 ,

    • [ 315 ] = 3 x 3 x 5 x 7 ,
    • [ 351 ] = 3 x 3 x 3 x 13 ,
    • [ 375 ] = 3 x 5 x 5 x 5 ,
    • [ 441 ] = 3 x 3 x 7 x 7 ,
    • [ 459 ] = 3 x 3 x 3 x 17 ,

    • [ 495 ] = 3 x 3 x 5 x 11 ,
    • [ 513 ] = 3 x 3 x 3 x 19 ,
    • [ 525 ] = 3 x 5 x 5 x 7 ,
    • [ 585 ] = 3 x 3 x 5 x 13 ,
    • [ 621 ] = 3 x 3 x 3 x 23 ,

    • [ 625 ] = 5 x 5 x 5 x 5 ,
    • [ 693 ] = 3 x 3 x 7 x 11 ,

    And multiplying these [ 17 Category-IV numbers in { Set D } ] by [ 27 ] will then yield the 17 [ desired numbers ] for { Category VII } .

• Very simply , { Category II } and { higher-order categoires } can be inter-linked .

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A Conjecture for further thoughts :

• We now make this conjecture for further thoughts :
  • Let [ N ] be a { positive integer greater than or equal to [ 6 ] } , and :
    • we set-up the value [ V ] equal to [ '3' raised to the N-th power ] .

    Let us now count-up , by { categories } , all the { odd numbers } up to the value [ V ] .

  • The Conjecture here is that :
    • { Category [ N ] } will have [ 1 member ] ,
    • { Category [ N - 1] } will have [ 4 members ] , and
    • { Category [ N - 2 ] } will have [ 17 members ] .

• Is the Conjecture right or is the Conjecture Wrong ?
  • More on this in the next Section .

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go to the next section : Section VIII - A First Proposal on Counting { Odd Numbers }

go to the last section : Section VI - Details on { Category III Odd-Numbers }

return to the { Counting Odd-Numbers } HomePage

original dated 2008-10-04