An Approach to Counting Odd Numbers

by Frank C. Fung - 1st published in October, 2008.

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Section V - Details on { Category II Odd Numbers }

Summary of the Section :

• In this Section , we shall provide further details on { Category II Odd-Numbers } .

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{ Category II Odd-Numbers } up to [ 19,683 ] :

• Let us first bring-in again the statistics for this category , from Secton IV :

  Category of Odd Number	Number of members in category	* * *	Smallest member in category	Largest member in category	Largest member in category factorized
  Category II	3,798		9	19,679	11 x 1,789

• Let us start-off by setting up a sufficiently-large { 2-Dimensional Matrix } , say of size [ 1,000 x 1,000 ] :
  • so that the value of the [ element ] at { row [ i ] , column [ j ] } is given by :
    • 

      where [ OP-i ] and [ OP-j ] are { Odd Prime Numbers } .

  We note here that :

  • For [ i ] = [ j ] :
    • 

      and this is an { element } on the diagonal of the matrix .

  • For [ i ] not equal to [ j ] , we have :
    • 

      so that the { 2-Dimensional Matrix } is symmetric and we need only to look at :

      • [ elements ] on the diagonal , and
      • [ elements ] to-the-top-right of the diagonal ;

        for further determinations .

• Let us now try to determine , for inclusion into our [ count ] , as to which of these [ elements ] are less than or equal to [ 19,683 ] ,
  • going on a row-by-row basis , as per this table below :

    X		Y		Value of E = X*Y	Count N = j - i + 1	* * *	Q = Next Prime greater than [ Y ]	Value of F = X*Q
    Value of [ X ]	Odd-Prime designation [ OP-i ]	Value of [ Y ]	Odd-Prime designation [ OP-j ]
    3	OP-1	6,553	OP-846	19,659	846		6,563	19,689
    5	OP-2	3,931	OP-545	19,655	544		3,943	19,715
    7	OP-3	2,803	OP-408	19,621	406		2,819	19,733
    11	OP-4	1,789	OP-277	19,679	274		1,801	19,811
    13	OP-5	1,511	OP-239	19,643	235		1,523	19,779
    17	OP-6	1,153	OP-190	19,601	185		1,163	19,771
    19	OP-7	1,033	OP-173	19,627	167		1,039	19,741
    23	OP-8	853	OP-146	19,619	139		857	19,711
    29	OP-9	677	OP-122	19,633	114		683	19,807
    31	OP-10	631	OP-114	19,561	105		641	19,871
    37	OP-11	523	OP-98	19,351	88		541	20,017
    41	OP-12	479	OP-91	19,639	80		487	19,967
    43	OP-13	457	OP-87	19,651	75		461	19,823
    47	OP-14	409	OP-79	19,223	66		419	19,693
    53	OP-15	367	OP-72	19,451	58		373	19,769
    59	OP-16	331	OP-66	19,529	51		337	19,883
    61	OP-17	317	OP-65	19,337	49		331	20,191
    67	OP-18	293	OP-61	19,631	44		307	20,569
    71	OP-19	277	OP-58	19,667	40		281	19,951
    73	OP-20	269	OP-56	19,637	37		271	19,783
    79	OP-21	241	OP-52	19,039	32		251	19,829
    83	OP-22	233	OP-50	19,339	29		239	19,837
    89	OP-23	211	OP-46	18,779	24		223	19,847
    97	OP-24	199	OP-45	19,303	22		211	20,467
    101	OP-25	193	OP-43	19,493	19		197	19,897
    103	OP-26	191	OP-42	19,673	17		193	19,879
    107	OP-27	181	OP-41	19,367	15		191	20,437
    109	OP-28	179	OP-40	19,511	13		181	19,729
    113	OP-29	173	OP-39	19,549	11		179	20,227
    127	OP-30	151	OP-35	19,117	6		157	19,939
    131	OP-31	149	OP-34	19,519	4		151	19,781
    137	OP-32	139	OP-33	19,043	2		149	20,413
    139	OP-33	139	OP-33	19,321	1		149	20,711
    TOTAL					3,798

    We note here that , in this determination process , we have gone :

    • [ 33 rows ] down , and
    • [ 846 columns ] across ,

      in our original { 2-Dimensional Matrix } of size [ 1,000 x 1,000 ] .

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Two Quick Comments :

• Comment #1 :

  The { square-root } of [ 19,683 ] is roughly [ 140.30 ] :

  • [ OP-33 ] at [ 139 ] is the { Odd-Prime } immediately below [ 140.3 ] , and
  • [ OP-34 ] at [ 149 ] is the { Odd-Prime } immediately above [ 140.3 ] .

  This then determines how many [ rows ] we need to go-down on the { Matrix } before stopping .

• Comment #2 :

  The value of [ '3' raised to the 8th power ] is [ 6,561 ] :

  • [ OP-846 ] at [ 6,553 ] is the { Odd-Prime } immediately below [ 6,561 ] , and
  • [ OP-847 ] at [ 6,563 ] is the { Odd-Prime } immediately above [ 6,561 ] .

  This then determines how many [ columns ] we need to go-across on the { Matrix } before stopping .

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• And we shall leave this , as-is-where-is , for the reader's further thoughts , for the moment .

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go to the next section : Section VI - Details on { Category III Odd-Numbers }

go to the last section : Section IV - Counting { odd numbers } up to [ 19,683 ]

return to the { Counting Odd-Numbers } HomePage

original dated 2008-10-04