For N to be divisible by n, but not by any integer less than n, N must be prime or a power. Therefore, the three consecutive numbers must be a combination of primes and powers (x^y).
Given three consecutive integers, one is divisible by 3. Therefore, one of the consecutive numbers must be a power of the form (3x)^y.
We need to consider consecutive integers of the forms:
power, power, power
power, power, prime
power, prime, power
prime, power, power (7, 8, 9 is an example)
prime, power, prime
That's as far as I can go. My guess is k=13, and the consecutive numbers are 7, 8, and 9.