On Counting Natural Numbers up to [ N = 1543 ]

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

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Section XI - Counting { Category VII Numbers } up to [ N ]

Summary for the Section :

• In this Section , we shall go thru the actual counting process for { Category VII 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 VII numbers } here ,

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 2 x 2 x 2 x 2 x 2 x 2 x [ 24.109375 ]

      involving [ N ] , the { 6th-power-of-[ 2 ] } , and their quotient [ 24.11 ] .

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

  • that :
    • [ 23 ] is the { next [ prime number ] smaller than [ 24.11 ] } , and
    • [ 29 ] is the { next [ prime number ] greater than [ 24.11 ] } .

• Consequently , we can now write :
  • [ 1472 ] = 2 x 2 x 2 x 2 x 2 x 2 x [ 23 ]
  • [ 1856 ] = 2 x 2 x 2 x 2 x 2 x 2 x [ 29 ]

  And we take note here that :

  • 1472 < 1543 < 1856

  And what this tells us here then is that :

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

      will involve a [ prime factor ] greater than the prime-number [ 23 ] .

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Setting the Dividing Lines :

• Let us recall that the seventh-root of [ 1543 ] is roughly [ 2.85 ] .

  We can then establish :

  • [ 2 ] is the next [ prime number ] smaller than [ 2.85 ] , and
  • [ 3 ] is the next [ prime number ] greater than [ 2.85 ] .

  Consequently , we can now sub-divide the { Category VII numbers } in-question here into six (6) sub-groups :

  • Type A :
    • These then are { [ Category VII numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • all 7 prime-factors being less-than-or-equal-to [ 2 ] .

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

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The Counting Procedure for { Category VII Numbers } :

• Let us now do the counting of { Category VII numbers } in six (6) stages :
  • Stage I :
    • Counting [ Type A ] and [ Type B ] { Category VII numbers } in one-shot ;
      • and this is done as a matter of convenience .

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

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

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

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

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

• STAGE I :
  • For this stage , we simply list-out the { Category VII numbers } fitting the description , as follows :
    • [ 128 ] = [ 2 x 2 x 2 x 2 x 2 x 2 x 2 ]

    • [ 192 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 3 ]
    • [ 320 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 5 ]
    • [ 448 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 7 ]
    • [ 704 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 11 ]
    • [ 832 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 13 ]
    • [ 1088 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 17 ]
    • [ 1216 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 19 ]
    • [ 1472 ] = [ 2 x 2 x 2 x 2 x 2 x 2 ] x [ 23 ]

    The count here for Stage I is therefore [ 9 ] .

• STAGE II :
  • For this stage , we simply list-out the { Category VII numbers } fitting the description , as follows :
    • [ 288 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 3 x 3 ]
    • [ 480 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 3 x 5 ]
    • [ 672 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 3 x 7 ]
    • [ 800 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 5 x 5 ]
    • [ 1056 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 3 x 11 ]
    • [ 1120 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 5 x 7 ]
    • [ 1248 ] = [ 2 x 2 x 2 x 2 x 2 ] x [ 3 x 13 ]

    The count here for Stage II is therefore [ 7 ] .

• STAGE III :
  • For this stage , we simply list-out the { Category VII numbers } fitting the description , as follows :
    • [ 432 ] = [ 2 x 2 x 2 x 2 ] x [ 3 x 3 x 3 ]
    • [ 720 ] = [ 2 x 2 x 2 x 2 ] x [ 3 x 3 x 5 ]
    • [ 1008 ] = [ 2 x 2 x 2 x 2 ] x [ 3 x 3 x 7 ]
    • [ 1200 ] = [ 2 x 2 x 2 x 2 ] x [ 3 x 5 x 5 ]

    The count here for Stage III is therefore [ 4 ] .

• STAGE IV :
  • For this stage , we simply list-out the { Category VII numbers } fitting the description , as follows :
    • [ 648 ] = [ 2 x 2 x 2 ] x [ 3 x 3 x 3 x 3 ]
    • [ 1080 ] = [ 2 x 2 x 2 ] x [ 3 x 3 x 3 x 5 ]
    • [ 1512 ] = [ 2 x 2 x 2 ] x [ 3 x 3 x 3 x 7 ]

    The count here for Stage IV is therefore [ 3 ] .

• STAGE V :
  • For this stage , we simply list-out the { Category VII numbers } fitting the description , as follows :
    • [ 972 ] = [ 2 x 2 ] x [ 3 x 3 x 3 x 3 x 3 ]

    The count here for Stage V is therefore [ 1 ] .

• STAGE VI :
  • For this stage , we simply list-out the { Category VII numbers } fitting the description , as follows :

    • [ 1458 ] = [ 2 ] x [ 3 x 3 x 3 x 3 x 3 x 3 ]

      The count here for Stage VI is therefore [ 1 ] .

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Total Count for { Category VII } :

• The Total Count of { Category VII Numbers } up to [ N = 1543 ] is therefore :
  • [ 25 ] = 9 + 7 + 4 + 3 + 1 + 1

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go to the next section : Section XII - Counting { Category VIII Numbers } up to [ N ]

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

return to the Counting Natural Numbers HomePage

Original dated 2012-10-08