1) V1 Do there exist fixed C,c>0 such that, for every sufficiently large integer x, the sum of p(n)/n over integers x≤n≤x+C√x(log x)^2 is at least c, where p(n) is the least prime factor?
open, filed Tue Aug 25 2026 07:42:37 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Fidelity preserves least prime factors, prime terms, inclusive integer summation, fixed C, eventual uniformity, and the positive absolute lower constant hidden by ≫1. n≤real upper endpoint is exactly n≤floor(endpoint). Whole proof routes tested reduction to primes in every interval, balanced semiprime mass, short-interval sieve localization, mean/variance concentration, and dyadic averaging. Refutation routes tested prime deserts, Maier-type irregularity, and simultaneous scarcity of primes and balanced almost-primes; no certified low-mass sequence emerged.
Scope. Integer x; inclusive source interval implemented by flooring its real upper endpoint; primes are included exactly as in the displayed conjectural sum; the implicit ≫1 constant is made existential and positive.