Do quasiperfect numbers exist? Search for $n$ with $\sigma(n)=2n+1$, or extend the exclusion bound (Guy UPINT §B2)
Statement
A quasiperfect number is a positive integer $n$ with $\sigma(n)=2n+1$ — equivalently, its proper divisors sum to $n+1$ (an abundance of exactly one). None is known. Exhibit a quasiperfect number, or verify that none exists up to a stated bound while strengthening the known constraints.
Acceptance. FULLY RESOLVES, either: (a) a quasiperfect $n$ with a prime factorization so that $\sigma(n)$ is machine-checkable and $\sigma(n)=2n+1$ is verified (would be the first known); or (b) a rigorous nonexistence proof — a complete finite/analytic argument, ideally machine-checkable. ADVANCES: extend the exclusion — certify that no quasiperfect $n$ exists with $\omega(n)\in\{7,8,\dots\}$ below a new bound $B>10^{35}$ via a reproducible search over odd squares of bounded prime-count, or tighten a structural constraint (e.g. raise the $\ge 7$ distinct-prime bound) with a checkable computation. Deliver the certificate and verifier.
Background
Guy, 'Unsolved Problems in Number Theory' (3rd ed.), §B2 (almost-perfect, quasi-perfect, and related numbers). No quasiperfect number has ever been found. Any quasiperfect number must be an odd perfect square, must exceed $10^{35}$, and must have at least seven distinct prime factors (Cattaneo; Hagis–Cohen; Abbott–Aull–Brown–Suryanarayana). The attack is a structured search over odd squares with bounded number of prime factors, verifying $\sigma(n)=2n+1$ from factorizations. Status note: the problem is still listed as open by standard references (Wikipedia; Wolfram MathWorld) and by the DeepMind formal-conjectures project (open issue #2239, as of 2026). An unrefereed 2025 preprint ('A Note on Quasiperfect Numbers', preprints.org 202506.2159) claims a nonexistence proof; it is not peer-reviewed and does not settle the question — a rigorous (ideally formalized) proof, or a witness, is what would resolve it.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Quasiperfect number — Wikipedia | website |
| REF-02 | Quasiperfect Number — Wolfram MathWorld | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.