I've completed searching for twins up to n=10K for k=1 to 100K. The list is attached. There are only 23 twins between n=5K and 10K. I can see that I'm going to need to expand the list up to k=1M or 10M to get any significant # of twins for n>10K. (no surprise there!)

I also checked the list for triplets and quadruplets. The largest of all 3 kinds that I've found so far are:

Twins: 33891*2^9869-1,+1

Triplets: 32811*2^2707-1,+1,+5

Quads: 3741*2^153-1,+1,+5,+7

I also checked triplets and quads for the form of k*2^n-7,-5,-1,+1 and k*2^n-5,-1,+1 but there were none as large.

-7, -5, +5, and +7 primes were checked at

http://www.alpertron.com.ar/ECM.HTM. The largest ones were also checked with Primo software.

Although the list looks small now, it only takes an exponent of 10475 to make the top-10 triplets list and an exponent of 3489 to make the top-10 quads list. Largest k-tuplets are shown at

www.ltkz.demon.co.uk/ktuplets.htm.

Gary