On Counting Natural Numbers up to [ N = 1543 ]

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

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

Summary for the Section :

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

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 2 x 2 x 2 x 2 x 2 x [ 48.21875 ]

      involving [ N ] , the { 5th-power-of-[ 2 ] } , and their quotient [ 48.22 ] .

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

  • that :
    • [ 47 ] is the { next [ prime number ] smaller than [ 48.22 ] } , and
    • [ 53 ] is the { next [ prime number ] greater than [ 48.22 ] } .

• Consequently , we can now write :
  • [ 1504 ] = 2 x 2 x 2 x 2 x 2 x [ 47 ]
  • [ 1696 ] = 2 x 2 x 2 x 2 x 2 x [ 53 ]

  And we take note here that :

  • 1504 < 1543 < 1696

  And what this tells us here then is that :

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

      will involve a [ prime factor ] greater than the prime-number [ 47 ] .

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Setting the Dividing Lines :

• Let us recall that the sixth-root of [ 1543 ] is roughly [ 3.40 ] .

  We can then establish :

  • [ 3 ] is the next [ prime number ] smaller than [ 3.40 ] , and
  • [ 5 ] is the next [ prime number ] greater than [ 3.40 ] .

  Consequently , we can now sub-divide the { Category VI numbers } in-question here into five (5) sub-groups :

  • Type A :
    • These then are { [ Category VI numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • all 6 prime-factors being less-than-or-equal-to [ 3 ] .

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

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

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

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

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

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The Counting Procedure for { Category VI Numbers } :

• Let us now do the counting of { Category VI numbers } in four (5) stages :
  • Stage I :
    • Counting [ Type A ] and [ Type B ] { Category VI numbers } in one-shot ;
      • and this is done as a matter of convenience .

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

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

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

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

• STAGE I :

  We then set up this table below for the actual count :

  Category V Number to be investigated		* * *	Value of [ N ] divided by [ X ]	* * *	Next Prime smaller than [ Z ]		* * *	Next Prime larger than [ Z ]		* * *	Control Value for Counting Purposes		* * *	Roving Count T = j - i + 1
  Value of Cat. V Number	Factors of Cat. V Number				Value of Prime	Rank of Prime		Value of Prime	Rank of Prime		Largest Prime Factor of [ X ]	Rank of Prime
  ( X )		*	( Z )	*	( P-j )	( j )	*	( P-k )	( k )	*	( P-i )	( i )	*	( T )
  32	2 x 2 x 2 x 2 x 2	***	48.22	***	47	15	***	53	16	***	2	1	***	15
  48	2 x 2 x 2 x 2 x 3	***	32.15	***	31	11	***	37	12	***	3	2	***	10
  72	2 x 2 x 2 x 3 x 3	***	21.43	***	19	8	***	23	9	***	3	2	***	7
  108	2 x 2 x 3 x 3 x 3	***	14.29	***	13	6	***	17	7	***	3	2	***	5
  162	2 x 3 x 3 x 3 x 3	***	9.52	***	7	4	***	11	5	***	3	2	***	3
  243	3 x 3 x 3 x 3 x 3	***	6.35	***	5	3	***	7	4	***	3	2	***	2
  *************	*************	***	********	***	********	********	***	********	********	***	********	********	***	********
  TOTAL														42

• STAGE II :
  • For this stage , we simply list-out the { Category VI numbers } fitting the description here , as follows :
    • [ 400 ] = [ 2 x 2 x 2 x 2 ] x [ 5 x 5 ]
    • [ 560 ] = [ 2 x 2 x 2 x 2 ] x [ 5 x 7 ]
    • [ 880 ] = [ 2 x 2 x 2 x 2 ] x [ 5 x 11 ]

    • [ 1040 ] = [ 2 x 2 x 2 x 2 ] x [ 5 x 13 ]
    • [ 1360 ] = [ 2 x 2 x 2 x 2 ] x [ 5 x 17 ]
    • [ 1520 ] = [ 2 x 2 x 2 x 2 ] x [ 5 x 19 ]

    • [ 784 ] = [ 2 x 2 x 2 x 2 ] x [ 7 x 7 ]
    • [ 1232 ] = [ 2 x 2 x 2 x 2 ] x [ 7 x 11 ]
    • [ 1456 ] = [ 2 x 2 x 2 x 2 ] x [ 7 x 13 ]

    • [ 600 ] = [ 2 x 2 x 2 x 3 ] x [ 5 x 5 ]
    • [ 840 ] = [ 2 x 2 x 2 x 3 ] x [ 5 x 7 ]
    • [ 1320 ] = [ 2 x 2 x 2 x 3 ] x [ 5 x 11 ]

    • [ 1176 ] = [ 2 x 2 x 2 x 3 ] x [ 7 x 7 ]

    • [ 900 ] = [ 2 x 2 x 3 x 3 ] x [ 5 x 5 ]
    • [ 1260 ] = [ 2 x 2 x 3 x 3 ] x [ 5 x 7 ]

    • [ 1350 ] = [ 2 x 3 x 3 x 3 ] x [ 5 x 5 ]

    The Total Count for Stage II here is therefore [ 16 ] .

• STAGE III :
  • For this stage , we simply list-out the { Category VI numbers } fitting the description here , as follows :
    • [ 1000 ] = [ 2 x 2 x 2 ] x [ 5 x 5 x 5 ]
    • [ 1400 ] = [ 2 x 2 x 2 ] x [ 5 x 5 x 7 ]

    • [ 1500 ] = [ 2 x 2 x 3 ] x [ 5 x 5 x 5 ]

  • The Total Count for Stage III here is therefore [ 3 ] .

• STAGE IV :
  • For this stage , our finding here is that there isn't a { Category VI numbers } fitting the description here .

    We note here that :

    • [ 2500 ] = [ 2 x 2 ] x [ 5 x 5 x 5 x 5 ]

      and this is already greater than [ N = 1543 ] .

    The Total Count for Stage IV is therefore [ 0 ] .

• STAGE V :
  • For this stage , our finding here is that there isn't a { Category VI numbers } fitting the description here .

    We note here that :

    • [ 6500 ] = [ 2 ] x [ 5 x 5 x 5 x 5 x 5 ]

      and this is already greater than [ N = 1543 ] .

    The Total Count for Stage V is therefore [ 0 ] .

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Total Count for Category VI :

• The Total Count of { Category VI numbers } up to [ N = 1543 ] is therefore :
  • [ 61 ] = 42 + 16 + 3 + 0 + 0

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

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

return to the Counting Natural Numbers HomePage

Original dated 2012-10-08