Your approach is interesting. AFAIK when selecting k's nobody tried to take into account the fact that LLR can process small k's much faster than larger ones. Possibly because the first version of LLR released in 2003 couldn't. Also when selecting, or better to say constructing k's (before checking the weight) it was customary (if I'm not mistaken) to select k so that for no n, k*2^n1 (or +1) is divisible by any small primes (smaller than a certain value) while 8331405*2^n1 is divisible by 11 and 13 for some values of n.
BTW, do you mind if I try k=25935 from 183,500 to about 210,000.
And congrats on many primes already found for k=8331405.
