That is true and I have actually checked the numbers of factors with different k's.

First there is no new examples where k=c*p, so the factor is 2*c*p^2 + 1, still only the one example: p=93077, c=674487.

Here is a list of numbers of factors of each k-value up to k=1024, and a graph of some of the values:

mersennefactors1.txt mersennefactors1.png
For k=2 (mod 4) there is no factors since for those k-values 2kp+1 is not +/- 1 (mod 8) for any prime p.

The number of factors goes down with higher k, but for each k=12*n there seems to be a local peak. The lowest number of factors seems to be at the prime k-values.

Here is a list of all the factors ordered by the length of k in binary, so that is factors where 2

^{n-1} <= k < 2

^{n}: (note that n=1 is just k=1, n=2: k=3, n=3: k=4,5,7)

mersennefactors2.txt
All the missing n-values have 0 factors of that length.