n => 2n+1 to get prime: seed = 73

[זקיר סעידוב - ד\"ר/Zakir Seidov Ph.D.]

Dear Seqfans,
The operation n => 2n+1 quickly gives primes for most "seed" values of n.
But for some seeds, the transformed numbers keep being composite.
The first "tough" number is n=73.
Here is the list of least divisors of transformed numbers
(I do not consider "seed" itself which in this case happened to be prime ):
{3,5,3,7,
3,5,3,19,
3,5,3,47,
3,5,3,7,
3,5,3,61,
3,5,3,29,
3,5,3,7,
3,5,3,1439,
3,5,3,73,
3,5,3,7,
3,5,3,19,
3,5,3,46703,
3,5,3,7,
3,5,3,22247,
3,5,3,59,
3,5,3,7,
3,5,3,761,
3,5,3,73,
3,5,3,7,
3,5,3,19,
3,5,3,137,
3,5,3,7,
3,5,3,131381,
3,5,3,2411639,
3,5,3,7}.
The clear pattern is seen, which may help to search the prime case.
My request is:
Can the n =>2n+1 transformation, in this particular case,
 lead to prime number (and when?),
may anyone bother to find it?
What about general theory?

Thank you very much,
Zak

PS  The last considered number, 
93806144416888975710756037197823,
has the full list of divisors as follows:
{{7, 1}, {13, 1}, {599, 1}, {21214924397, 1}, {81118812239619751, 1}} -
at least according to Mathematica.
And next three numbers are clearly composite according the pattern mentioned.

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