A Conjecture on the Collatz-Kakutani Path Length for the Mersenne Primes
  We present here a new conjecture for the nature of the Mersenne prime numbers by connecting it with the Collatz-Kakutani problem. By introducing a natural path length on the basis of the Collatz-Kakutani tree, we conjecture that this path length of a Mersenne prime from the root of the Collatz-Kakutani tree is approximately proportional to the index of the Mersenne prime. We also discuss difference of behaviors between Mersenne numbers and Mersenne primes. 
Toru Ohira and Hiroshi Watanabe
math.NT