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 have looked a Charlie's suggested approach to proving that 4 in a row is not possible.

I can proved that one cannot have 4 consecutive integers of the form [m,4p,n,2q^2] or [4p,n,2q^2,m] where p and q are prime. This would lead to 4p = 2q^2 -2

which simplifies to 2p = (q-1)(q+1) which has no prime solutions.

Thus, 6 in a row is impossible.

4 in a row would also be impossible if we could prove that one cannot have 4 consecutive integers of the form [n,2q^2,m,4p] or [2q^2,m,4p,n] where p and q are prime. This leads to 2p = q^2+1, which is obviously not impossible.

Solutions include (p,q) = (5,3), (13,5), and (61,11)

So, I have not yet proved that 4 is impossible

*Edited on ***April 2, 2017, 1:57 pm**