sw00877math--1-3-5-7-9

 Certain groups with 5 terms.  Not sure where this will go.

1 3 5 7 9   3 9 15 21 27   5 15 25 35 45   7 21 35 49 63   9 27 45 63 81

Two times in the above is the only odd integer prime triad: 3 5 7

11 33 55 77 99   13 39 65 91 117   15 45 75 105 135   17 51 85 119 153
       19 57 95 133 171   21 63 105 147 189   23 69 115 161 207

Not much to behold.   11 13  17 19  23

So what is of interest?  x 1  x 3  x 5  x 7  x 9

The first integer in each group gets the group name, but not the gpo.
A group's position is always its (first integer + 1)/2.  Thus, the gpo
of the 23 Group is 12.  It happens that "23" is a prime number and
that therefore 24 (3 x 8) - 1 equals a prime number.  So what?  I
am not certain, but (1 x 8) - 1 equals a prime number too.  What
other single-digit integers work?  (9 x 8) - 1, but neither (5 x 8) - 1
nor (7 x 8) - 1.  (2 x 8) - 1?  No.  (4 x 8) - 1?  Yes.  (6 x 8) - 1?
Yes.  (8 x 8) - 1?  No.  Messy.  Yet maybe not.  The "Yes" ones
are: 7, 23, 31, 47, 71.  The "No" ones are: 15, 39, 55, 63.  It
appears that multiples of "3" matter extensively.

Something else.  The what's missing game.  This involves powns only
after 3 5 7, and forget integers that have "5" for a final digit.  In the 3
Group the missing powns are 11, 13, 17, 19, 23.  Beyond this group
the search toughens/ because some of the missing integers show up in
higher groups.  My interest in this method is weakening.

Something other.  The spreads, and where they occur.  Of course, a
higher strangeness exists in the higher realms, so I may not get far here
either.  3 [2] 5 [2] 7 [4] 11 [2] 13 [4] 17 [2] 19 [4] 23 [6] 29 [2] 31 [6]
37 [4] 41 [2] 43 [4] 47 [6] 53 [6] 59 [2] 61 [6] 67 [4] 71 [2] 73 [6] 79
[4] 83 [6] 89 [8] 97 [4] 101 [2] 103 [4] 107 [2] 109 [4] 113 [14] 127 
Notice that before we encounter the two prime pairs above 100 we hit
the first 8-apart, and after 113 we hit a stunning 14-apart dead zone.
What's with 115, 117, 119 121, 123, 125?  5, 3, 7, 11, 3, 5.  "119",
being two less than an odd square, should be (according to previous
experiences) a prime number, but 7 x 17 invalidates that assumption.
The immediate realization is that more of the same can be expected. 

083
085   5 x 17
087   3 x 29
089
091   7 x 13
093   3 x 31
095   5 x 19
097
099   3 x 3 x 11
101   
103
105   3 x 5 x 7   (105 is a major node integer)
107
109
111   3 x 37
113
115   5 x 23
117   3 x 3 x 13
119   7 x 17
121   11 x 11
123   3 x 41
125   5 x 5 x 5
127





#
Brian A. J. Salchert