On Counting Natural Numbers up to [ N = 1543 ]

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

Top Of Page

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

Summary for the Section :

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

go to Top Of Page

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 III numbers } here ,

  • let us now write-out this specific arithmetic relation :
    • [ 1543 ] = 2 x 2 x [ 385.75 ]

      involving [ N ] , the { square-of-[ 2 ] } , and their quotient [ 385.75 ] .

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

  • that :
    • [ 383 ] is the { next [ prime number ] smaller than [ 385.75 ] } , and
    • [ 389 ] is the { next [ prime number ] greater than [ 385.75 ] } .

• Consequently , we can now write :
  • [ 1532 ] = 2 x 2 x [ 383 ]
  • [ 1556 ] = 2 x 2 x [ 389 ]

  And we take note here that :

  • 1532 < 1543 < 1556

  And what this tells us here then is that :

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

      will involve a [ prime factor ] greater than the prime-number [ 383 ] .

go to Top Of Page

Setting the Dividing Lines :

• Let us recall that the cube-root of [ 1543 ] is roughly [ 11.56 ] .

  We can then establish :

  • [ 11 ] is the next [ prime number ] smaller than [ 11.56 ] , and
  • [ 13 ] is the next [ prime number ] greater than [ 11.56 ] .

  Consequently , we can now sub-divide the { Category III numbers } in-question here into three (3) sub-groups :

  • Type A :
    • These then are { [ Category III numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • all 3 prime-factors being less-than-or-equal-to [ 11 ] .

  • Type B :
    • These then are { [ Category III numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 2 prime-factors being less-than-or-equal-to [ 11 ] , and
      • 1 prime-factor being greater-than-or-equal-to [ 13 ] .

  • Type C :
    • These then are { [ Category III numbers ] less-than-or-equal-to [ N = 1543 ] } with :
      • 1 prime-factor being less-than-or-equal-to [ 11 ] , and
      • 2 prime-factors being greater-than-or-equal-to [ 13 ] .

go to Top Of Page

The Counting Procedure for { Category III Numbers } :

• Let us now do the counting of { Category III numbers } in two (2) stages :
  • Stage I :
    • Counting [ Type A ] and [ Type B ] { Category III numbers } in one-shot ;
      • and this is done as a matter of convenience .

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

• STAGE I :

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

  Category II 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. II Number	Factors of Cat. II 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 )
  4	2 x 2	***	385.75	***	383	76	***	389	77	***	2	1	***	76
  6	2 x 3	***	257.17	***	257	55	***	263	56	***	3	2	***	54
  9	3 x 3	***	171.44	***	167	39	***	173	40	***	3	2	***	38
  10	2 x 5	***	154.30	***	151	36	***	157	37	***	5	3	***	34
  15	3 x 5	***	102.87	***	101	26	***	103	27	***	5	3	***	24
  25	5 x 5	***	61.72	***	61	18	***	67	19	***	5	3	***	16
  14	2 x 7	***	110.21	***	109	29	***	113	30	***	7	4	***	26
  21	3 x 7	***	73.48	***	73	21	***	79	22	***	7	4	***	18
  35	5 x 7	***	44.09	***	43	14	***	47	15	***	7	4	***	11
  49	7 x 7	***	31.49	***	31	11	***	37	12	***	7	4	***	8
  22	2 x 11	***	70.14	***	67	19	***	71	20	***	11	5	***	15
  33	3 x 11	***	46.75	***	43	14	***	47	15	***	11	5	***	10
  55	5 x 11	***	28.05	***	23	9	***	29	10	***	11	5	***	5
  77	7 x 11	***	20.04	***	19	8	***	23	9	***	11	5	***	4
  121	11 x 11	***	12.75	***	11	5	***	13	6	***	11	5	***	1
  **********	**********	***	********	***	********	********	***	********	********	***	********	********	***	********
  TOTAL														340

• STAGE II :

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

  Category II 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. II Number	Factors of Cat. II Number				Value of Prime	Rank of Prime		Value of Prime	Rank of Prime
  ( X )		*	( Z )	*	( P-j )	( j )	*	( P-k )	( k )	*	( T )
  169	13 x 13	***	9.13	***	7	4	***	11	5	***	4
  221	13 x 17	***	6.98	***	5	3	***	7	4	***	3
  247	13 x 19	***	6.25	***	5	3	***	7	4	***	3
  299	13 x 23	***	5.16	***	5	3	***	7	4	***	3
  377	13 x 29	***	4.09	***	3	2	***	5	3	***	2
  403	13 x 31	***	3.83	***	3	2	***	5	3	***	2
  481	13 x 37	***	3.21	***	3	2	***	5	3	***	2
  533	13 x 41	***	2.89	***	2	1	***	3	2	***	1
  559	13 x 43	***	2.76	***	2	1	***	3	2	***	1
  611	13 x 47	***	2.53	***	2	1	***	3	2	***	1
  689	13 x 53	***	2.24	***	2	1	***	3	2	***	1
  767	13 x 59	***	2.01	***	2	1	***	3	2	***	1
  289	17 x 17	***	5.34	***	5	3	***	7	4	***	3
  323	17 x 19	***	4.78	***	3	2	***	5	3	***	2
  391	17 x 23	***	3.95	***	3	2	***	5	3	***	2
  493	17 x 29	***	3.13	***	3	2	***	5	3	***	2
  527	17 x 31	***	2.93	***	2	1	***	3	2	***	1
  629	17 x 37	***	2.45	***	2	1	***	3	2	***	1
  697	17 x 41	***	2.21	***	2	1	***	3	2	***	1
  731	17 x 43	***	2.11	***	2	1	***	3	2	***	1
  361	19 x 19	***	4.27	***	3	2	***	5	3	***	2
  437	19 x 23	***	3.53	***	3	2	***	5	3	***	2
  551	19 x 29	***	2.80	***	2	1	***	3	2	***	1
  589	19 x 31	***	2.62	***	2	1	***	3	2	***	1
  703	19 x 37	***	2.19	***	2	1	***	3	2	***	1
  529	23 x 23	***	2.92	***	2	1	***	3	2	***	1
  667	23 x 29	***	2.31	***	2	1	***	3	2	***	1
  713	23 x 31	***	2.16	***	2	1	***	3	2	***	1
  **********	**********	***	********	***	********	********	***	********	********	***	********
  TOTAL											47

  Two special comments here on this Table above :

  • Our primary targets of investigation here , as per the very-left-column in the above Table , are then :
    • { Category II numbers } with both prime-factors being greater-than-or-equal-to [ 13 ] .

  • For the [ 1st Row ] in the Table above :
    • [ 169 = 13 x 13 ] is the above-said { Category II Number } with both prime-factors being greater-than-or-equal-to-[ 13 ] .

    And the 4 { Category III Numbers } that made-up the [ count-of-4 ] for this [ 1st Row ] are then :

    • 13 x 13 x 2 = [ 338 ] ,
    • 13 x 13 x 3 = [ 507 ] ,
    • 13 x 13 x 5 = [ 845 ] , and
    • 13 x 13 x 7 = [ 1183 ] .

go to Top Of Page

Total Count for Category III :

• The Total Count of { Category III Numbers } up to [ N = 1543 ] is therefore :
  • [ 387 ] = 340 + 47

go to Top Of Page

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

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

return to the Counting Natural Numbers HomePage

Original dated 2012-10-05