Density of Mersenne divisors

[Random Poster]

Suppose we have n consecutive primes starting with p; out of these, we would expect n*(some function of p) to divide a Mersenne number with prime exponent. What is known about this function?

[R.D. Silverman]

Originally Posted by Random Poster

Suppose we have n consecutive primes starting with p; out of these, we would expect n*(some function of p) to divide a Mersenne number with prime exponent. What is known about this function?

A lot is known.

q = 2p+1 will divide M_p if q is a q.r. of p This happens 1/2 the time.
q = 2kp + 1 will divide M_p if q is a '2k'th power residue of p. When
and how frequently this happens is determined by Artin's higher reciprocity
laws.

It is an exercize for a grad student in analytic number theory to
estimate the function you seek.....