603, 604, and 605 are the first 3 consecutive integers that are the product of a prime and another prime squared.

603=3^{2}*67

604=2^{2}*151

605=5*11^{2}

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.

A030515 (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.