On Counting Natural Numbers up to [ N = 1543 ]

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

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Section VIII - Counting { Category IV Numbers } up to [ N ]

Summary for the Section :

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

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 2 x 2 x 2 x [ 192.875 ]

      involving [ N ] , the { cube-of-[ 2 ] } , and their quotient [ 192.88 ] .

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

  • that :
    • [ 191 ] is the { next [ prime number ] smaller than [ 192.88 ] } , and
    • [ 193 ] is the { next [ prime number ] greater than [ 192.88 ] } .

• Consequently , we can now write :
  • [ 1528 ] = 2 x 2 x 2 x [ 191 ]
  • [ 1544 ] = 2 x 2 x 2 x [ 193 ]

  And we take note here that :

  • 1528 < 1543 < 1544

  And what this tells us here then is that :

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

      will involve a [ prime factor ] greater than the prime-number [ 191 ] .

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Setting the Dividing Lines :

• Let us recall that the fourth-root of [ 1543 ] is roughly [ 6.27 ] .

  We can then establish :

  • [ 5 ] is the next [ prime number ] smaller than [ 6.27 ] , and
  • [ 7 ] is the next [ prime number ] greater than [ 6.27 ] .

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

  • Type A :
    • These then are { [ Category IV numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • all 4 prime-factors being less-than-or-equal-to [ 5 ] .

  • Type B :
    • These then are { [ Category IV numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 3 prime-factors being less-than-or-equal-to [ 5 ] , and
      • 1 prime-factor being greater-than-or-equal-to [ 7 ] .

  • Type C :
    • These then are { [ Category IV numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 2 prime-factors being less-than-or-equal-to [ 5 ] , and
      • 2 prime-factors being greater-than-or-equal-to [ 7 ] .

  • Type D :
    • These then are { [ Category IV numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 1 prime-factor being less-than-or-equal-to [ 5 ] , and
      • 3 prime-factors being greater-than-or-equal-to [ 7 ] .

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The Counting Procedure for { Category IV Numbers } :

• Let us now do the counting of { Category IV numbers } in three (3) stages :
  • Stage I :
    • Counting [ Type A ] and [ Type B ] { Category IV numbers } in one-shot ;
      • and this is done as a matter of convenience .

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

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

• STAGE I :

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

  Category III 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. III Number	Factors of Cat. III 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 )
  8	2 x 2 x 2	***	192.88	***	191	43	***	193	44	***	2	1	***	43
  12	2 x 2 x 3	***	128.58	***	127	31	***	137	32	***	3	2	***	30
  18	2 x 3 x 3	***	85.72	***	83	23	***	89	24	***	3	2	***	22
  27	3 x 3 x 3	***	57.15	***	53	16	***	59	17	***	3	2	***	15
  20	2 x 2 x 5	***	77.15	***	73	21	***	79	22	***	5	3	***	19
  30	2 x 3 x 5	***	51.43	***	47	15	***	53	16	***	5	3	***	13
  50	2 x 5 x 5	***	30.86	***	29	10	***	31	11	***	5	3	***	8
  45	3 x 3 x 5	***	34.29	***	31	11	***	37	12	***	5	3	***	9
  75	3 x 5 x 5	***	20.57	***	19	8	***	23	9	***	5	3	***	6
  125	5 x 5 x 5	***	12.34	***	11	5	***	13	6	***	5	3	***	3
  **********	**********	***	********	***	********	********	***	********	********	***	********	********	***	********
  TOTAL														168

• STAGE II :
  • For Stage II , let us first write-out this arithmetic relation :
    • 1543 = 2 x 2 x [ 385.75 ]
      • involving [ N ] , the { square-of-[ 2 ] } and their quotient [ 385.75 ] .

    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 [ 7 ]
      • that are less than the quotient [ 385.75 ] .

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

    	7	11	13	17	19	23	29	31	37	41	43	47	53	59
    7	49	77	91	119	133	161	203	217	259	287	301	329	371	413
    11		121	143	187	209	253	319	341	407	451	473	517	583	649
    13			169	221	247	299	377	403	481	533	559	611	689	767
    17				289	323	391	493	527	629	697	731	799	901	1003
    19					361	437	551	589	703	779	817	893	1007	1121
    23						529	667	899	851	943	989	1081	1219	1357
    ****	****	****	****	****	****	****	****	****	****	****	****	****	****	****

    And the 28 { 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 II Number to be investigated		* * *	Value of [ N ] divided by [ X ]	* * *	Category II Number under consideration ( with both prime factors being [ 7 ] or above )																												* * *	Roving Count T
  Value of Cat. II Number	Factors of Cat. II Number				4 9	7 7	9 1	1 1 9	1 2 1	1 3 3	1 4 3	1 6 1	1 6 9	1 8 7	2 0 3	2 0 9	2 1 7	2 2 1	2 4 7	2 5 3	2 5 9	2 8 7	2 8 9	2 9 9	3 0 1	3 1 9	3 2 3	3 2 9	3 4 1	3 6 1	3 7 1	3 7 7
  ( X )		*	( Z )	*					*					^					*					^					*				*	( T )
  4	2 x 2	*	385.75	*	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	*	28
  6	2 x 3	*	257.17	*	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	N	N	N	N	N	N	N	N	N	N	N	N	*	16
  9	3 x 3	*	171.44	*	Y	Y	Y	Y	Y	Y	Y	Y	Y	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	*	9
  10	2 x 5	*	154.30	*	Y	Y	Y	Y	Y	Y	Y	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	*	7
  15	3 x 5	*	102.87	*	Y	Y	Y	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	*	3
  25	5 x 5	*	61.72	*	Y	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	N	*	1
  ********	********	*	********	*	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	**	*	*******
  TOTAL																																		64

• STAGE III :

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

  Category III Number to be investigated		* * *	Value of [ N ] divided by [ X ]	* * *	Next Prime smaller than [ Z ]		* * *	Next Prime larger than [ Z ]		* * *	Roving Count T = j
  Value of Cat. III Number	Factors of Cat. III Number				Value of Prime	Rank of Prime		Value of Prime	Rank of Prime
  ( X )		*	( Z )	*	( P-j )	( j )	*	( P-k )	( k )	*	( T )
  343	7 x 7 x 7	***	4.50	***	3	2	***	5	3	***	2
  539	7 x 7 x 11	***	2.86	***	2	1	***	3	2	***	1
  637	7 x 7 x 13	***	2.40	***	2	1	***	3	2	***	1
  **********	**********	***	********	***	********	********	***	********	********	***	********
  TOTAL											4

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Total Count for Category IV :

• The Total Count of { Category IV Numbers } up to [ N = 1543 ] is therefore :
  • [ 236 ] = 168 + 64 + 4

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

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

return to the Counting Natural Numbers HomePage

Original dated 2012-10-08