On Counting Natural Numbers up to [ N = 1543 ]

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

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Section IX - Counting { Category V Numbers } up to [ N ]

Summary for the Section :

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

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Setting-up the Upper-Limit for our investigations here :

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

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

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 2 x 2 x 2 x 2 x [ 96.4375 ]

      involving [ N ] , the { 4th-power-of-[ 2 ] } , and their quotient [ 96.44 ] .

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

  • that :
    • [ 89 ] is the { next [ prime number ] smaller than [ 96.44 ] } , and
    • [ 97 ] is the { next [ prime number ] greater than [ 96.44 ] } .

• Consequently , we can now write :
  • [ 1424 ] = 2 x 2 x 2 x 2 x [ 89 ]
  • [ 1552 ] = 2 x 2 x 2 x 2 x [ 97 ]

  And we take note here that :

  • 1424 < 1543 < 1552

  And what this tells us here then is that :

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

      will involve a [ prime factor ] greater than the prime-number [ 89 ] .

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Setting the Dividing Lines :

• Let us recall that the fifth-root of [ 1543 ] is roughly [ 4.34 ] .

  We can then establish :

  • [ 3 ] is the next [ prime number ] smaller than [ 4.34 ] , and
  • [ 5 ] is the next [ prime number ] greater than [ 4.34 ] .

  Consequently , we can now sub-divide the { Category V numbers } in-question here into four (4) sub-groups :

  • Type A :
    • These then are { [ Category V numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • all 5 prime-factors being less-than-or-equal-to [ 3 ] .

  • Type B :
    • These then are { [ Category V numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 4 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 V numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 3 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 V numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 2 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 V 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 V Numbers } :

• Let us now do the counting of { Category V numbers } in four (4) stages :
  • Stage I :
    • Counting [ Type A ] and [ Type B ] { Category V numbers } in one-shot ;
      • and this is done as a matter of convenience .

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

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

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

• STAGE I :

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

  Category IV 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. IV Number	Factors of Cat. IV 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 )
  16	2 x 2 x 2 x 2	***	96.44	***	89	24	***	97	25	***	2	1	***	24
  24	2 x 2 x 2 x 3	***	64.29	***	61	18	***	67	19	***	3	2	***	17
  36	2 x 2 x 3 x 3	***	42.86	***	41	13	***	43	14	***	3	2	***	12
  54	2 x 3 x 3 x 3	***	28.57	***	23	9	***	29	10	***	3	2	***	8
  81	3 x 3 x 3 x 3	***	19.05	***	19	8	***	23	9	***	3	2	***	7
  **********	**********	***	********	***	********	********	***	********	********	***	********	********	***	********
  TOTAL														68

• STAGE II :
  • For Stage II , let us first write-out this arithmetic relation :
    • 1543 = 2 x 2 x 2 x [ 192.875 ]
      • involving [ N ] , the { cube-of-[ 2 ] } and their quotient [ 192.88 ] .

    In order to proceed with the counting here , it is now necessary for us to identify :

    • { Category II numbers } with both prime-factors being greater-than-or-equal-to [ 5 ]
      • that are less than the quotient [ 192.88 ] .

    We then set-up this Multiplication Table below specifically for this purpose :

    	5	7	11	13	17	19	23	29	31	37	41
    5	25	35	55	65	85	95	115	145	155	185	205
    7		49	77	91	119	133	161	203	217	259	287
    11			121	143	187	209	253	319	341	407	451
    13				169	221	247	299	377	403	481	533
    17					289	323	391	493	527	629	697
    ****	****	****	****	****	****	****	****	****	****	****	****

    And the 20 { Category II Numbers } fitting the description are then as marked-off in { blue-color } in the Table above .

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

  Category III Number to be investigated		* * *	Value of [ N ] divided by [ X ]	* * *	Category II Number under consideration ( with both prime factors being [ 5 ] or above )																				* * *	Roving Count T
  Value of Cat. III Number	Factors of Cat. III Number				2 5	3 5	4 9	5 5	6 5	7 7	8 5	9 1	9 5	1 1 5	1 1 9	1 2 1	1 3 3	1 4 3	1 4 5	1 5 5	1 6 1	1 6 9	1 8 5	1 8 7
  ( X )		*	( Z )	*					*					^					*					^	*	( T )
  8	2 x 2 x 2	*	192.88	*	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	*	20
  12	2 x 2 x 3	*	128.58	*	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	N	N	N	N	N	N	N	N	*	12
  18	2 x 3 x 3	*	85.72	*	Y	Y	Y	Y	Y	Y	Y	N	N	N	N	N	N	N	N	N	N	N	N	N	*	7
  27	3 x 3 x 3	*	57.14	*	Y	Y	Y	Y	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	*	4
  **********	**********	*	********	*	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	*	*******
  TOTAL																										43

• STAGE III :
  • For this stage , we simply list-out the { Category V numbers } fitting the description , as follows :
    • [ 500 ] = [ 2 x 2 ] x [ 5 x 5 x 5 ]
    • [ 750 ] = [ 2 x 3 ] x [ 5 x 5 x 5 ]
    • [ 1125 ] = [ 3 x 3 ] x [ 5 x 5 x 5 ]

    • [ 700 ] = [ 2 x 2 ] x [ 5 x 5 x 7 ]
    • [ 1050 ] = [ 2 x 3 ] x [ 5 x 5 x 7 ]

    • [ 1100 ] = [ 2 x 2 ] x [ 5 x 5 x 11 ]

    • [ 1300 ] = [ 2 x 2 ] x [ 5 x 5 x 13 ]

    • [ 980 ] = [ 2 x 2 ] x [ 5 x 7 x 7 ]
    • [ 1470 ] = [ 2 x 3 ] x [ 5 x 7 x 7 ]

    • [ 1540 ] = [ 2 x 2 ] x [ 5 x 7 x 11 ]

    • [ 1372 ] = [ 2 x 2 ] x [ 7 x 7 x 7 ]

    The Total Count for Stage III is therefore [ 11 ] .

• STAGE IV :
  • For this stage , we simply list-out the { Category V number } fitting the description , as follows :
    • [ 1250 ] = [ 2 ] x [ 5 x 5 x 5 x 5 ]

    The Total Count for Stage IV is therefore [ 1 ] .

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Total Count for Category V :

• The Total Count of { Category V Numbers } up to [ N = 1543 ] is therefore :
  • [ 123 ] = 68 + 43 + 11 + 1

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

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

return to the Counting Natural Numbers HomePage

Original dated 2012-10-08