ProofAtlas

Divisor sums and additive number theory

Existence of odd weird numbers

Collaboration beta

An odd weird number would have more than enough proper divisors in total to exceed the number, yet no selection of distinct proper divisors would add up to it. This handoff supplies restrictions and construction equations, but no such number or universal exclusion.

∃n∈ℤ>0odd:σ(n)>2n,n∉B(n),B(n)={∑d∈Ud:U⊆{d:d∣n, d<n}}.
Erdős problem #470 — odd weird numbers question
Known results and sources
Illustration: The deficient integer 315 has proper-divisor total 309, with two holes at 2 and 307 in its subset-sum interval; it is not weird.
Illustration: Missing subset sums occur for 315, but its proper-divisor sum is too small for abundance; it is not a weird number.

Research problem

Exact mathematical statement

For a positive integer n, let σ(n) be the sum of all its positive divisors. An integer is abundant when σ(n)>2n, and semiperfect when a subset of its distinct proper positive divisors sums to n. A weird integer is abundant and not semiperfect.

Decide whether

∃n∈ℤ>0,n≡1(mod2),σ(n)>2n,n∉{∑d∈Ud:U⊆{d∈ℤ>0:d∣n, d<n}}.\exists n\in\mathbb Z_{>0},\quad n\equiv1\pmod2,\quad \sigma(n)>2n,\quad n\notin\left\{\sum_{d\in U}d: U\subseteq\{d\in\mathbb Z_{>0}:d\mid n,\ d<n\}\right\}.

Each proper divisor can be used at most once. A positive resolution requires an actual integer or rigorously specified factorization, certified prime factors, exact abundance, and a proof of the subset-sum obstruction. A negative resolution must cover every odd abundant integer.

Problem infographic

Problem at a glance

Illustration: Odd weirdness requires abundance and failure of a distinct-divisor subset sum. In the deficient example 315, green ranges are attainable sums and gold cells mark the unattainable sums 2 and 307.
Illustration: For 315, green marks attainable subset sums and gold marks the holes 2 and 307; its proper-divisor total is only 309. An odd integer satisfying both weirdness conditions remains an open existence question.

Current mathematical picture

Where work on Existence of odd weird numbers stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureAssuming every deficient odd core has only the two universal gaps

The core 98595=315·313 has extra missing sums including 310 and 312, and a missing multiple 630 of gcd(m,σ(m))=3. The source's exact two-gap and progression theorems remain valid for cores that satisfy their hypotheses; the compatible excess 630 supplies a separate construction route.

Route status · Narrowed route
Main reductionA distinct sufficient construction equation

The source reports that 32656σ(t)−32865t=105 with every prime divisor of t greater than 630 would make n=98595t odd weird with excess 630. A squarefree multiplier needs at least seven new bases; the arbitrary-exponent argument gives only at least five. No admissible multiplier is established.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeRealize the excess-630 equation with an admissible multiplier, or prove a uniform obstruction across a specified surviving multiplier family.Task status · Ready to work on

Work mapped so far

Existence of odd weird numbers in numbers

1.2kretained lines of mathematical investigation1,166 in the current working snapshot
Argument development
666 · 57%
Explored or eliminated routes
74 · 6%
Computational analysis
79 · 7%
Open obligations
272 · 23%
Definitions and setup
75 · 6%
9selected mapped statements2routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedSep 7, 2026
Source recordedSep 7, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

14 selected steps

Scroll horizontally to explore the route

Working route overview for Existence of odd weird numbersA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can an odd abundant integer fail every proper-divisor subset sum? — Depends on missing premiseCan an odd abundant integerfail every proper-divisorsubset…A distinct sufficient construction equation — Depends on missing premiseA distinct sufficientconstruction equationCurrent reduction — Depends on missing premiseCurrent reductionExcess two is one sufficient route — Depends on missing premiseExcess two is one sufficientrouteA local eight-base exclusion — Depends on missing premiseA local eight-base exclusionClosing target — Depends on missing premiseClosing targetProgression coverage forces semiperfectness — Depends on missing premiseProgression coverage forcessemiperfectnessRestricted squarefree support bounds — Depends on missing premiseRestricted squarefreesupport boundsThree new bases, arbitrary exponents — Depends on missing premiseThree new bases, arbitraryexponentsAssuming every deficient odd core has only the two universal gaps — stoppedAssuming every deficient oddcore has only the twouniversal…Promoting a coprime-abundant witness to an odd weird number — stoppedPromoting a coprime-abundantwitness to an odd weirdnumberRealize the excess-630 equation with an admissible multiplier, or prove a uniform obstruction across a specified surviving multiplier family. — OpenRealize the excess-630equation with an admissiblemultiplier,…Remove finite caps on earlier-prime exponents in the excess-2 route without confusing the already eliminated terminal exponent with the remaining problem. — OpenRemove finite caps onearlier-prime exponents inthe…Control the first positive excess for every odd primitive abundant candidate beyond the special progression and two-gap core classes. — OpenControl the first positiveexcess for every oddprimitive…
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded
Narrowed routeAssuming every deficient odd core has only the two universal gaps

The core 98595=315·313 has extra missing sums including 310 and 312, and a missing multiple 630 of gcd(m,σ(m))=3. The source's exact two-gap and progression theorems remain valid for cores that satisfy their hypotheses; the compatible excess 630 supplies a separate construction route.

Route status · Narrowed route
Narrowed routePromoting a coprime-abundant witness to an odd weird number

The source's seven-base all-odd-exponent witness is abundant and coprime but has a certified distinct-proper-divisor representation of its excess, so it is semiperfect. Such examples still delimit arithmetic pruning; a construction must add a complete subset-sum obstruction rather than infer weirdness from coprimality.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Realize the excess-630 equation with an admissible multiplier, or prove a uniform obstruction across a specified surviving multiplier family.Suggested move: Suggested next step: Study five- and six-base nonsquarefree multipliers with every prime factor greater than 630 and no cap on the terminal exponent; retain exact divisibility, primality, and small-divisor conditions.
Ready to work on
02
Remove finite caps on earlier-prime exponents in the excess-2 route without confusing the already eliminated terminal exponent with the remaining problem.Suggested move: Suggested next step: Combine exact divisor-sum divisibility with rigorous bounds on exponent tails, and prove any partition into exponent residue classes complete, including exceptional primes and denominator cases.
Ready to work on
03
Control the first positive excess for every odd primitive abundant candidate beyond the special progression and two-gap core classes.Suggested move: Suggested next step: Combine deficiency constraints from every complete prime-power deletion and determine whether their joint restrictions force the relevant excess to be representable as a distinct-divisor subset sum.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusOpen problem

It remains unknown whether an odd weird number exists. Weird numbers are abundant but cannot be expressed as a sum of a subset of their proper divisors.

[2][1]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Computational resultFang reports a computation finding no odd weird numbers up to 10^21. The finite search does not settle existence.[2]
3 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusOdd weird numbers
Related problemInfinitely many primitive weird numbers

The same numbered problem also asks a distinct infinitude question about primitive weird numbers; that component is not the odd-number existence target.

[1]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • formal statement · statement onlyErdős 470 odd-weird-number existence statement

    The Lean file states the existence question using Weird and Odd. The answer and proof contain sorry placeholders, so this is not a proof of either answer.

    [3]
  • computation · not independently reproducedFang's odd-weird-number search

    The preprint describes a reverse-search computation and its finite exclusion result. No search implementation or output certificate was executed or checked here.

    [2]

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes
Statements by mathematical role9 selected mapped statements
  • theorem candidate1 of 91
  • reduction3 of 93
  • lemma3 of 93
  • computational claim2 of 92
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included
  • retained route statementCan an odd abundant integer fail every proper-divisor subset sum?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExcess two is one sufficient routeintermediate
  • retained route statementProgression coverage forces semiperfectnessintermediate
  • retained route statementThree new bases, arbitrary exponentsintermediate
  • retained route statementRestricted squarefree support boundsintermediate
  • retained route statementA local eight-base exclusionintermediate
  • retained route statementA distinct sufficient construction equationintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureAssuming every deficient odd core has only the two universal gapsreported failure
  • Useful failurePromoting a coprime-abundant witness to an odd weird numberreported failure
  • Research targetRealize the excess-630 equation with an admissible multiplier, or prove a uniform obstruction across a specified surviving multiplier family.open
  • Research targetRemove finite caps on earlier-prime exponents in the excess-2 route without confusing the already eliminated terminal exponent with the remaining problem.open
  • Research targetControl the first positive excess for every odd primitive abundant candidate beyond the special progression and two-gap core classes.open
  • ComputationThe source reports exact Python and C++ searches for restricted excess-2 families, with exact terminal elimination for the largest-prime exponent and explicit incomplete outcomes for unclosed tails or prime-table exhaustion.Reported squarefree excess-2 bounds are at least 25 bases, at least 60 when divisible by 3, and at least 62 when coprime to 15. Six-base solutions require a smaller-prime exponent at least 11; seven-base solutions require one at least 3. The computations were not independently replayed for this draft. · reported unreproduced
  • Narrowed routeAssuming every deficient odd core has only the two universal gapsThe core 98595=315·313 has extra missing sums including 310 and 312, and a missing multiple 630 of gcd(m,σ(m))=3. The source's exact two-gap and progression theorems remain valid for cores that satisfy their hypotheses; the compatible excess 630 supplies a separate construction route.
  • Narrowed routePromoting a coprime-abundant witness to an odd weird numberThe source's seven-base all-odd-exponent witness is abundant and coprime but has a certified distinct-proper-divisor representation of its excess, so it is semiperfect. Such examples still delimit arithmetic pruning; a construction must add a complete subset-sum obstruction rather than infer weirdness from coprimality.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeRealize the excess-630 equation with an admissible multiplier, or prove a uniform obstruction across a specified surviving multiplier family.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointRealize the excess-630 equation with an admissible multiplier, or prove a uniform obstruction across a specified surviving multiplier family.

Existence of odd weird numbers · ready to start

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Research contextPrepared context for any AI agent

An odd weird number would have more than enough proper divisors in total to exceed the number, yet no selection of distinct proper divisors would add up to it. This work supplies restrictions and construction equations, but no such number or universal exclusion.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Erdős Problem #470maintained problem list · accessed Sep 7, 2026
  2. 2
    Searching on the boundary of abundance for odd weird numberspreprint · Wenjie Fang · 2022-07-26 · ARXIV 2207.12906v1 · accessed Sep 7, 2026
  3. 3
    Formal Conjectures — Erdős Problem 470formalization · Google DeepMind Formal Conjectures · accessed Sep 7, 2026

Important qualifications

  • The bounded research did not establish the conjecture's original proposer, proposal date, or original-publication binding.
  • Direct access to the maintained Erdős problem page returned HTTP 403. Its indexed entry was inspected; Fang's primary preprint independently supports the open question.
  • The reported finite search was not reproduced. Its separate abundance-restricted extension is not treated as an unrestricted exclusion bound.
  • The public Lean statement contains placeholders and was not built or audited. This private draft grants no mathematical acceptance, rights, authorship, contributor credit, or publication authority.

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