Finally I hit a big one:

**21*2^999599-1** is prime! 310911 digits

currently the 80th largest of all times

After testing k=21 from 500k to 950k in increasing order I decided to try the top range from 1M to 995k in reversed order and the prime was right there on line 7 of the file!

Achieved limits of this and other k's (no other primes found):

k=7: 1.180 M

k=21: 950k

k=55: 430k

k=77: 700k

k=79: 520k stopped

k=89: 500k

Continuing k=55 and k=89 to 520k, k=21 to 1M, k=7 and k=77 until further notice