On shifted primes with large prime factors and their products

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n up to x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p-1 : p | n) exceeds n^{1/2k}.