[seqfan] A260119; Least positive integer k such that 2^n-1 and k^n-1 are relatively prime.

[David Corneth]

So I've put a conjecture that might help finding such values of k, but I
don't see a proof and I guess a faster program could be found. There are
some examples to it, but I don't know what characteristic these n have. The
conjecture is in the history of the sequence, see
https://oeis.org/history/view?seq=A260119&v=36

 Any ideas on how to proceed?