Jordan-Cototient Record Limit at Zero
Abstract
For each positive natural input, eventual strict Jordan-cototient record membership at sufficiently small positive parameters is characterized by the number of distinct prime factors, with three exceptional values.
Theorem 1.1 (The eventual record set near zero).
Proof. Machine-checked in Lean as D5/S3/Factorization/JordanCototientRecordLimitZero.a387335_eventual_record_iff (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every natural n >= 1, there is a positive real epsilon such that n is a strict Jordan-cototient record for every real parameter k with 0 < k < epsilon exactly when n is one, two, four, or has at least two distinct prime factors. Equivalently, after the three exceptional values, these eventual records are the non-prime-powers listed by OEIS A024619.
The threshold may depend on n. At parameter zero every input n > 1 has Jordan cototient one. Its right derivative is log(n) when n has at least two distinct prime factors, and log(p^(a-1)) when n=p^a. Strict derivative comparisons hold simultaneously for the finitely many positive predecessors of n.
For exclusion, an odd prime power p^a with a >= 2 loses to 2p^(a-1), while 2^a with a >= 3 loses to 32^(a-2). Every prime greater than two ties with two. These comparisons prove the pointwise eventual characterization stated in the OEIS A387335 comment of Hal M. Switkay dated 2025-11-30.