Patterns in A129783?

[Jon Schoenfield]

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Warning:
Ignorant newbie e-mail follows;
reader discretion is advised.    ;-)
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All,

I don't yet have a proof about any of the conjecturally-missing numbers in 
A129783, but I did get (after using the first 283 primes, i.e., the primes 
up through 1847) a list identical to the one currently given in the OEIS. 
The eight numbers that required the largest primes were all perfect squares:

n =  16 was first found at p =   89, q =   97
n =  25 was first found at p =  139, q =  149
n =  36 was first found at p =  199, q =  211
n =  81 was first found at p =  523, q =  541
n = 100 was first found at p =  887, q =  907
n = 121 was first found at p = 1129, q = 1151
n = 144 was first found at p = 1669, q = 1693
n =  64 was first found at p = 1831, q = 1847

In each case, n = ((q - p) / 2)^2 , so p * q + n = ((p + q) / 2)^2.

That much kinda made sense to me ....  But I'm curious about some of the 
other patterns I've noticed.  In particular, in looking at the numbers that 
do and (conjecturally) don't occur in A129783, I'm wondering if there's some 
simple explanation as to why the list seems to include so many numbers that 
are congruent to 26 mod 36.  Among all the integers in the range from 278 
through 926, barely a third are in A129783, but every integer congruent to 
26 mod 36 in that range is in A129783.

In the 1000 characters on the lines below (36 per line, except the last), 
the X's represent numbers in A129783 (starting with 1), and the periods 
represent numbers that (conjecturally) aren't in the sequence.  If the lines 
are displayed in a fixed-pitch font (e.g., Courier New), the density of X's 
in column 26 seems striking (to me, anyway <g>)!

X.XX....XX...X.X..X.X.X.XX..XX...XXX
.X....XX.XX.X...X....X...X.XXXXX....
..X.XX..XXX.XX.....X.XX....X..X..X..
XX..X.X.XXX.XX..X...X....X...XXX..XX
.X.XX....X......XXXX..X.X...XX....X.
XX.....X.X.....X......X..X...X.XX.X.
.XXXXX..X.X.....X...X.X.X...X.X..X..
XX.XX....XX.....X....X...X....XX....
XX.XXX..XX...X.....XX....X..XX....XX
.X...X.X..X.....X....XX..X..XXX.X...
XX.XX....X...X.....X....XX.....X.XX.
XX.X.X.X.XX.....X.....X..X..XXXX....
X.X.X...XX...XX.X..XX.X..X..X.X...X.
XXX.XX...XX..X.X....X....X..X.X.....
...X.....X...X....XX....XX..XX..X...
XX..X..XX.......XX..X.X..X...X.X...X
X..X..X..XX..X..XX....X..X..X.X..X..
....X.XX.XX.XX..X...X.X..X..X.X.X...
.X.XX....X..X......X.X...X.X.XX...X.
.......XXX...X..XX....X..X..XX..X.X.
.XX.X...X.......X..X..X..X..XX......
XX...X.X.XX.X...X..XXX...X.X..XXX...
.X..X.....X..X.....XX....X...XX..XX.
....X.X..XX.XX...........X....XX....
XX.X....X....X..XX.XX.X..X...XX..X.X
XX.....X.X......X.....X..X.....XX.X.
....XX...XX..X...XX.X...X...X...X.X.
X....X....X.....XX..........

If you copy all 1000 characters into Microsoft WordPad, make it all a 
fixed-pitch font, delete the carriage return/line feed at the end of each 
line to combine it all into one long line, and select View > Options... > 
Wrap to Window, then various patterns may seem to emerge as you drag the 
right edge of the window to widen and narrow it....

Okay, I figure that's considered a barbaric approach.  :-)  Are there any 
recommended tools/methods for detecting such patterns?

Thanks for bearing with another newbie email,

-- Jon