You can create the list of good multipliers "k" on your own easily and check if the numbers fit to your model. I can also send you a txt file with all the k in proper order if you want to.
I try to understand you model:
The sum of divisors of m is maximal if m is prime. So you expect prime numbers to be good "k"?
Or did you mean the number of divisors? Or number of different divisors?
Here are the multipliers "k" (in order) with leads to factorizations without extra multiplier m:
1155,105,165,330,210,255,285,273,15,390,21,345,561,495,195,231,435,462,315,385,33,510,420,30,715
Smooth numbers with exponent 1 are more likely yes. But I'm not able to predict the not so likely numbers.
