I was using srsieve_0.6.17 which has been the latest version since May 31, 2010 as far as I can tell.

Yes all these bases have one or two k being a perfect cube or square.

Does srsieve need some special flag to take account of perfect cubes or squares?

Or do I need to do another step before/after using srsieve?

For R463:

>srsieve -n 100001 -N 110000 -P 1e6 "216*463^n-1", I get 269 terms remaining

>srsieve -n 100001 -N 110000 -P 1e6 "356*463^n-1", I get 252 terms remaining

For R696:

>srsieve -n 100001 -N 110000 -P 1e6 "152*696^n-1", I get 705 terms remaining

>srsieve -n 100001 -N 110000 -P 1e6 "225*696^n-1", I get 1014 terms remaining

For R774:

>srsieve -n 100001 -N 110000 -P 1e6 "25*774^n-1", I get 671 terms remaining

>srsieve -n 100001 -N 110000 -P 1e6 "30*774^n-1", I get 447 terms remaining

For R588:

>srsieve -n 100001 -N 110000 -P 1e6 "3*588^n-1", I get 795 terms remaining

>srsieve -n 100001 -N 110000 -P 1e6 "16*588^n-1", I get 664 terms remaining

For R828:

>srsieve -n 100001 -N 110000 -P 1e6 "64*828^n-1", I get 404 terms remaining

>srsieve -n 100001 -N 110000 -P 1e6 "68*828^n-1", I get 676 terms remaining

For S140:

>srsieve -n 100001 -N 110000 -P 1e6 "8*140^n+1", I get 328 terms remaining

>srsieve -n 100001 -N 110000 -P 1e6 "16*140^n+1", I get 642 terms remaining

For S533:

>srsieve -n 100001 -N 110000 -P 1e6 "38*533^n+1", I get 747 terms remaining

>srsieve -n 100001 -N 110000 -P 1e6 "64*533^n+1", I get 691 terms remaining

The values I get are always higher, so it's consistent that more factors needing to be eliminated.

Keeping on top of all these conjectures is a really big task and you and others obviously put in a massive effort.

I am sure everyone appreciates it.