1) V1 Let F(n) be the supremum over natural k of omega(n+k) log(log k)/log k, where omega counts distinct prime factors.
open, filed Tue Aug 25 2026 07:38:41 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Then F(n) tends to infinity with n.
Whole proof route 1 chooses n+k highly smooth via primorial/CRT constructions; the natural choice k comparable to the smooth number yields only the known constant-scale lower bound because omega times loglog(k)/log(k) stays O(1). Route 2 seeks unusually smooth numbers in short intervals after n; available smooth-number gap estimates do not force the unbounded gain. Route 3 optimizes k through products of selected small primes and residue constraints, but controlling k relative to n+k is the unresolved core. Refutation would require a uniform upper bound on omega(n+k) relative to k for infinitely many n; no such covering obstruction is known.
Scope. Natural n and shifts k exactly as in the current source; real logarithms and the atTop-to-atTop divergence formulation.