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Untouchable Number

A positive integer lying outside the image of the aliquot-sum function, classified by the nonexistence of any positive integer whose proper divisors sum to it.

Version
v2 · 2026-09-06 · History
Domain-specific #
3042
Origin domain
mathematics
Subdomain
number theory
Aliases
Nonaliquot number, Non-aliquot number

Core Idea

An untouchable number is a positive integer that is not an aliquot sum of any positive integer. If \(\sigma(n)\) denotes the sum of all positive divisors of \(n\), define the sum-of-proper-divisors function by \(s(n)=\sigma(n)-n\). A positive integer \(m\) is untouchable exactly when there is no positive integer \(n\) satisfying \(s(n)=m\), or equivalently when \(m\notin s(\mathbb N_{>0})\). The term ‘nonaliquot number’ is standard in analytic number theory and makes the image-complement identity explicit.

Scope of Application

Untouchable numbers are used to study fibers, omitted values, density, and dynamics of the sum-of-proper-divisors function.

  • Arithmetic-function images. Determining which positive integers occur as \(s(n)\).
  • Inverse problems. Searching for or excluding preimages of a target aliquot sum.
  • Analytic number theory. Estimating the density and distribution of nonaliquot values.
  • Computational number theory. Enumerating certified initial ranges while distinguishing search limits from proofs.
  • Aliquot dynamics. Identifying vertices of the functional graph with indegree zero.
  • Historical number theory. Tracing divisor-sum classifications across medieval and modern terminology.

Clarity

Define \(s(n)\) and the positive-integer domain before listing examples. State whether \(s(1)=0\); this convention does not make zero part of the candidate class when untouchable numbers are defined as positive. To prove a value touchable, give a preimage and list or formula-check its proper divisors. To claim untouchability, do not substitute failure of a finite search for a proof unless the search comes with a justified preimage bound.

Manages Complexity

The abstraction turns an unbounded inverse search into a clean functional-graph and set-complement problem. Draw one directed edge \(n\to s(n)\) for each positive integer. A number is untouchable exactly when no edge enters it. This view separates witness-based positive results from global exclusions and reveals why iteration and preimage questions differ. Algebraic forms provide reusable certificates of touchability: \(p+1=s(p^2)\) for prime \(p\), and \(p+3=s(2p)\) for odd prime \(p\).

Abstract Reasoning

  1. Define \(s(n)=\sigma(n)-n\) on positive integers and fix the candidate value \(m\). 2. Search for structural preimage forms whose divisor sums can equal \(m\). 3. If a witness \(n\) is found, factor it and verify \(s(n)=m\) exactly. 4. If exclusion is sought, partition possible preimages by parity, factorization type, or divisor constraints. 5. Use identities such as \(s(p^2)=p+1\) to remove entire target families from the complement.

Knowledge Transfer

The strict parent is Complement: untouchable numbers are exactly everything in the declared universe of positive integers that is not in the designated subset given by the image of \(s\). The abstraction transfers a general inverse-image lesson: a property defined by missing outputs demands a domain, mapping, and image before its complement is meaningful. Analogues include nontotients and noncototients, but their arithmetic fibers and distribution theorems must be established separately.

Relationships to Other Abstractions

Local relationship map for Untouchable NumberParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Untouchable NumberDOMAINPrime abstraction: Complement — is a kind ofComplementPRIME

Current abstraction Untouchable Number Domain-specific

Parents (1) — more general patterns this builds on

  • Untouchable Number is a kind of Complement Prime

    Complement is the strict parent because the class is defined exactly as \(\mathbb N_{>0}\setminus s(\mathbb N_{>0})\).

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Untouchable Number sits in a sparse region of the domain-specific corpus (87th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Computational Number Theory & Enumeration (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08