After looking at some testing done by Willem, I've added another factor to the above pattern that holds with what has already been outlined in this thread as follows:

For all bases b where b == (40 mod 41), algebraic factors on even-n combine with a numeric factor on odd-n in the following scenario:

k=m^2 and m==(9 or 32 mod 41)

Factors to:

for even-n, let n=2q:

(m*b^q-1)*(m*b^q+1)

-and-

odd-n: factor of 41

Therefore, the above applies to bases 40, 81, 122, etc. for k=9^2, 32^2, 50^2, 73^2, etc.

In the case of Willem's Riesel base 40, this only affected k's that were m==(9 or 114 mod 123), which is a subset of m==(9 or 32 mod 41), because the others came were k==(1 mod 3), which has a trivial factor of 3 and so are not considered.

BTW, Willem correctly recognized such k's and eliminated them from his testing. Nice job Willem!

Although initially somewhat surprising that the factor of 41 as it applies to eliminating k's as a result of combining with algebraic factors had not been encountered up to this point, it is understandable. For b==(40 mod 41), base 40 has a huge conjecture and had not previously been started, bases 81 and 122 have small conjectures that are < than the smallest possible k that the above applies to, (k=81), base 163 with a conjecture of k=186 cannot have odd k's, which eliminates k=81, the only possible k that the above applies to, and bases 204, 245, 286, etc. have not yet been tested.

I would guess that we are down to the final few factors that haven't been found yet that this scenario applies to for bases <= 1024. The conjectures, as a general rule, get smaller as the bases get larger and so it becomes increasingly unlikely for any additional (mainly larger) factors to have an affect, generally because the smallest applicable k-value comes in above the conjecture or is eliminated by trivial factors.

Gary