On the values of the divisor function

For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p - 1) : p ≤ x prime} and {τ(2ⁿ - 1) : n ≤ x}.