On Counting Natural Numbers up to [ N = 1543 ]

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

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Summary and Key Findings :

• In this paper , we shall do a full count of all non-zero Natural Numbers up to [ 1543 ] ,
  • on a category-by-category basis .

  And the Categories here are classified as follows :

  • [ Category Zero ] shall contain the number [ 1 ] , this being the one-and-only element in this category ;

  • [ Category I ] shall contain all the [ prime numbers ] ;

  • [ Category II ] shall contain all [ composite numbers ] that are the product of 2 prime-numbers ;

  • [ Category III ] shall contain all [ composite numbers ] that are the product of 3 prime-numbers ;

  • And so-on-and-so-forth .

• And the Summary Table for the actual count is as follows :

  Category of Number	* * *	Smallest Number in Category		* * *	Largest Number in Category		* * *	Roving Count
  		Value of Number	Prime Factors of Number		Value of Number	Prime Factors of Number
  Category Zero	***	1	--	***	1	--	***	1
  Category I	***	2	2	***	1543	1543	***	243
  Category II	***	4	2 x 2	***	1541	23 x 67	***	450
  Category III	***	8	2 x 2 x 2	***	1542	2 x 3 x 257	***	387
  Category IV	***	16	2 x 2 x 2 x 2	***	1528	2 x 2 x 2 x 191	***	236
  Category V	***	32	2 x 2 x 2 x 2 x 2	***	1539	3 x 3 x 3 x 3 x 19	***	123
  Category VI	***	64	2 x 2 x 2 x 2 x 2 x 2	***	1504	2 x 2 x 2 x 2 x 2 x 47	***	61
  Category VII	***	128	2 x 2 x 2 x 2 x 2 x 2 x 2	***	1512	2 x 2 x 2 x 3 x 3 x 3 x 7	***	25
  Category VIII	***	256	2 x 2 x 2 x 2 x 2 x 2 x 2 x 2	***	1408	2 x 2 x 2 x 2 x 2 x 2 x 2 x 11	***	11
  Category IX	***	512	2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2	***	1280	2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5	***	4
  Category X	***	1024	2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2	***	1536	2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3	***	2
  **********	***	********	***************************	***	********	***************************	***	**********
  TOTAL								1543

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Table of Content for the paper :

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go to the 1st section : Section I - Introduction

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Original dated 2012-10-08