603, 604, and 605 are the first 3 consecutive integers that are the product of a prime and another prime squared.
1. What is the first set of 4 consecutive integers that are the product of a prime and another prime squared?
2. What is the first set of 5 consecutive integers that are the product of a prime and another prime squared?
I had a thought that another way to express this problem is to first think of the sequence of numbers of the form p*q^2: 12,18,20,28,44,45... Then the problem asks us to find runs consecutive integers in this sequence.
The OEIS does have this as sequence A054753
. They have a link to a stackexchange question
which asks and eventually answers the problem. The first set of five consecutive integers starts with 10093613546512321.
(numbers with exactly 6 factors) is a superset of A054753 and has some more information relevant to the problem. Specifically a reference to A141621
which is the sequence of the runs of five consecutive numbers in A030515/A054753.