sw00295math-nnssmagic.numbers

[ last modified:  2008-10-20 ] 

Natural Number Summation Sequence Magic Numbers

 2  8  18  32  50  72  98  128  162  200  242  288  338  392  450  n

 these numbers relate to nnss pairs 

 [ 04-04-07 insert:  for those it aids:
   -
   The natural number summation sequence (nnss) consists of
   whole numbers derived from all the positive whole numbers.
   All the positive whole numbers include zero.  So:
   0 + 1 = 1, making "1" the first sum.  1 + 2 = 3, making "3"
   the second sum.  These two numbers constitute the first
   nnss pair.  They are both odd positive whole numbers,
   and "2" is the magic number between them.  In the nnss
   each new sum is a member of a pair of either odd sums
   or even sums, and each of these sums is either the "< run"
   (< = less than) sum or the "> run" (> = greater than) sum
   in its pair, and each sum is a term in the nnss.  Odd nnss 
   pairs alternate with even nnss pairs; therefore, the pair  
   following the "1" / "3" pair is going to be an even pair.  
   Okay: 3 + 3 = 6, and 4 + 6 = 10.  The "magic" number
   between 6 and 10 is "8", wherein 8 - 2 = 6 and 8 + 2 = 10
   (which is the sum of 0 + 1 + 2 + 3 + 4).  In my reckoning
   I assign what I call a "termposition" (tpo) to each pair, and
   I use this tpo for deriving the nnss terms in that pair.  Do I
   need to know in advance a given pair of nnss terms?  No.
   However, this:  The first termposition here is "1" and each
   succeeding termposition is the next greater positive whole
   number.  Also, if a termpositon is an odd positive whole
   number, the nnss terms derivable from it will be "opwn"
   values.  If a termpostion is an even positive whole number,
   the nnss terms derivable from it will be "epwn" values.  See
   below for formulas.  Given this fact, any tpo from "1" into
   forever can be chosen to discover a pair of nnss terms, the
   greater of which is the sum of all the positive whole numbers
   through that "pwn" which is equal to the tpochosen times 2.
   In my "in memory of Gauss" example below, 5050 is the sum
   of its 50 tpo times 2 (the sum of all the consecutive positive 
   whole numbers through 100).  My apologies to those for whom
   this information in unnecessary.   Brian A. J. Salchert ]

 #

 tpo = termposition                 tpo x 2tpo = nnss magic number
 -
 2tpo = spread between the nnss terms at its tpo 
 -
 -
 At tpo 50 the nnss m n = 5000
 and the "< run" nnss term = 4950
 and the "> run" nnss term = 5050  (in memory of Gauss) 

 #

 tpo 1             2             2 - 1 = 1              2 + 1 = 3

 tpo 2             8             8 - 2 = 6              8 + 2 = 10

 tpo 3           18            18 - 3 = 15          18 + 3 = 21

 tpo 4           32            32 - 4 = 28          32 + 4 = 36

 tpo 5           50            50 - 5 = 45          50 + 5 = 55

 tpo 6           72            72 - 6 = 66          72 + 6 = 78

 tpo 7           98            98 - 7 = 91          98 + 7 = 105



 see number theory entry
 -           

 nnssmagic

 salcherts termposition mathematics

 bajs math entry 6

 
see for Wolfram MathWorld Natural Number recommended terminology


  

#
Brian A. J. Salchert