On Counting Natural Numbers up to [ N = 1543 ]

by Frank C. Fung ( 1st published in October, 2012. )

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Section XIV - Counting { Category X Numbers } up to [ N ]

Summary for the Section :

• In this Section , we shall go thru the actual counting process for { Category X Numbers } up to [ N ] ,
  • with [ N ] here being the natural number [ 1543 ] .

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Setting up the Upper-Limit for our investigation in this Section :

• Let us recall that [ 2 ] is the smallest [ prime number ] .

  And , for our purpose of counting { Category X numbers } here ,

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 [ 3.01371875 ]

      involving [ N ] , the { 9th-power-of-[ 2 ] } , and their quotient [ 3.02 ] .

  We can then quickly establish , from our [ Table of Prime Numbers ] ,

  • that :
    • [ 3 ] is the { next [ prime number ] smaller than [ 3.02 ] } , and
    • [ 5 ] is the { next [ prime number ] greater than [ 3.02 ] } .

• Consequently , we can now write :
  • [ 1536 ] = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x [ 3 ]
  • [ 2560 ] = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x [ 5 ]

    and also :

  • [ 2304 ] = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x [ 3 ] x [ 3 ]

  And we take note here that :

  • 1536 < 1543 < 2304

    and

  • 1536 < 1543 < 2560

  And what this tells us here then is that :

  • it will be most unlikely that :
    • any { [ Category X number ] less-than-or-equal-to [ N = 1543 ] }

      will involve a [ prime factor ] greater than the prime-number [ 3 ] .

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Setting the Dividing Lines :

• Let us recall that the tenth-root of [ 1543 ] is roughly [ 2.08 ] .

  We can then establish :

  • [ 2 ] is the next [ prime number ] smaller than [ 2.08 ] , and
  • [ 3 ] is the next [ prime number ] greater than [ 2.08 ] .

  Consequently , we can now sub-divide the { Category X numbers } in-question here into ten (10) sub-groups :

  • Type A :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • all 10 prime-factors being less-than-or-equal-to [ 2 ] .

  • Type B :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 9 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 1 prime-factor being greater-than-or-equal-to [ 3 ] .

  • Type C :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 8 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 2 prime-factors being greater-than-or-equal-to [ 3 ] .

  • Type D :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 7 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 3 prime-factors being greater-than-or-equal-to [ 3 ] .

  • Type E :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 6 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 4 prime-factors being greater-than-or-equal-to [ 3 ] .

  • Type F :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 5 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 5 prime-factors being greater-than-or-equal-to [ 3 ] .

  • Type G :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 4 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 6 prime-factors being greater-than-or-equal-to [ 3 ] .

  • Type H :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 3 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 7 prime-factors being greater-than-or-equal-to [ 3 ] .

  • Type J :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 2 prime-factors being less-than-or-equal-to [ 2 ] , and
      • 8 prime-factors being greater-than-or-equal-to [ 3 ] .

  • Type K :
    • These then are { [ Category X numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 1 prime-factor being less-than-or-equal-to [ 2 ] , and
      • 9 prime-factors being greater-than-or-equal-to [ 3 ] .

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The Counting Procedure for { Category X Numbers } :

• Let us now do the counting of { Category X numbers } in nine (9) stages :
  • Stage I :
    • Counting [ Type A ] and [ Type B ] { Category X numbers } in one-shot ;
      • and this is done as a matter of convenience .

  • Stage II :
    • Counting [ Type C ] { Category X numbers } .

  • Stage III :
    • Counting [ Type D ] { Category X numbers } .

  • Stage IV :
    • Counting [ Type E ] { Category X numbers } .

  • Stage V :
    • Counting [ Type F ] { Category X numbers } .

  • Stage VI :
    • Counting [ Type G ] { Category X numbers } .

  • Stage VII :
    • Counting [ Type H ] { Category X numbers } .

  • Stage VIII :
    • Counting [ Type J ] { Category X numbers } .

  • Stage IX :
    • Counting [ Type K ] { Category X numbers } .

• STAGE I :
  • For this stage , we simply list-out the { Category X numbers } fitting the description , as follows :
    • [ 1024 ] = [ 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ]

    • [ 1536 ] = [ 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ] x [ 3 ]

    The count here for Stage I is therefore [ 2 ] .

• STAGE II thru STAGE IX :
  • For these 8 stages , our finding here is that there simply isn't a { Category X number } fitting the description here .

    We note here that :

    • [ 2304 ] = [ 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ] x [ 3 x 3 ]

    • [ 3456 ] = [ 2 x 2 x 2 x 2 x 2 x 2 x 2 ] x [ 3 x 3 x 3 ]

    • [ 5184 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 3 x 3 x 3 x 3 ]

    • [ 7776 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 3 x 3 x 3 x 3 x 3 ]

    • [ 11664 ] = [ 2 x 2 x 2 x 2 ] x [ 3 x 3 x 3 x 3 x 3 x 3 ]

    • [ 17496 ] = [ 2 x 2 x 2 ] x [ 3 x 3 x 3 x 3 x 3 x 3 x 3 ]

    • [ 26244 ] = [ 2 x 2 ] x [ 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 ]

    • [ 39366 ] = [ 2 ] x [ 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 ]

      and all 8 numbers above are already greater than [ N = 1543 ] .

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Total Count for { Category X } :

• The Total Count of { Category X Numbers } up to [ N = 1543 ] is therefore [ 2 ] .

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go to the next section : Section XV - Counting Summary for all Catagories

go to the last section : Section XIII - Counting { Category IX Numbers } up to [ X ]

return to the Counting Natural Numbers HomePage

Original dated 2012-10-08