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.^[1]

This is a preimage-exclusion property, not a statement about the divisors of \(m\) alone. For example, \(4\) is touchable because the proper divisors of \(9\) are \(1\) and \(3\), so \(s(9)=4\). The number \(6\) is touchable because \(s(6)=6\); perfect numbers are therefore automatically in the image. In contrast, \(2\) and \(5\) are untouchable. Historical work reported by Sesiano traces recognition of \(2\) and \(5\) to medieval Islamic number theory.^[2] The modern sequence begins \(2,5,52,88,96,120,124,146,\ldots\), but sequence membership is defined by nonexistence of a preimage, not by position in a table.

The family supports both elementary exclusions and deep distribution questions. If \(p\) is prime, then \(s(p^2)=1+p\), so no number of the form \(p+1\) is untouchable. If \(p\) is odd prime, \(s(2p)=1+2+p=p+3\), so numbers three above an odd prime are touchable. Erdős proved that untouchable numbers are not sporadic but form an infinite, indeed positive-proportion, phenomenon.^[3] Chen and Zhao later proved that at least \(0.06x+o(x)\) positive integers up to \(x\) are nonaliquot.^[1] It remains unproved whether \(5\) is the only odd untouchable number; heuristic and Goldbach-type reasoning must be labeled as conditional or conjectural rather than mixed into the definition.

Structural Signature¶

The positive-integer universe. Both candidate values and possible preimages lie in a declared positive domain.
The divisor-sum function. \(\sigma(n)\) sums all positive divisors of \(n\).
The aliquot-sum map. \(s(n)=\sigma(n)-n\) removes the number itself and sums its proper divisors.
The image. Touchable values are exactly those attained by \(s\) on a positive integer.
The complement predicate. Untouchability means membership in \(\mathbb N_{>0}\setminus s(\mathbb N_{>0})\).
The witness obligation. To show touchability, one explicit \(n\) with \(s(n)=m\) suffices.
The exclusion obligation. To prove untouchability, all possible preimages must be ruled out by theorem or exhaustive bounded reasoning.
The parity boundary. Even values dominate known examples while the uniqueness of odd example \(5\) remains open.
The density question. Counting nonaliquot values concerns image distribution, not merely computing initial terms.
The terminology boundary. Aliquot sequence dynamics concerns iterating \(s\); untouchability concerns whether a value has any incoming edge.

What It Is Not¶

Not a number with no proper divisors. Primes have proper divisor \(1\), but the predicate concerns being someone else's divisor sum.
Not a perfect number. A perfect number satisfies \(s(n)=n\) and is therefore touchable.
Not a weird number. Weirdness concerns abundant numbers not expressible as sums of distinct proper divisors.
Not a nontotient. Nontotients lie outside the image of Euler's totient function, a different map.
Not an aliquot sequence. An aliquot sequence follows repeated forward images rather than missing preimages.
Not proof that all other odd numbers are touchable. The odd classification remains an open problem.

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. Separate known theorems from conjectures: infinitely many untouchables and positive lower density are proven, while ‘5 is the only odd untouchable’ is not. Distinguish natural density, lower density, and a finite observed proportion. The phrase ‘sum of proper divisors of any number’ includes the possibility that the preimage equals the target, which is why perfect numbers are touchable. Use ‘nonaliquot’ as an alias only in this image-complement sense, since ‘aliquot’ elsewhere can denote a proper divisor or an iterated sequence.

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 \(p\). Such identities eliminate large structured families without searching individual divisor lists. Analytic methods then count the complement of the image rather than enumerating it one term at a time. Computational tables remain useful for conjecture and calibration, but the abstraction keeps their epistemic status visible. A program may certify the initial sequence within a bounded domain if it proves a sufficient preimage search bound; a list generated by an arbitrary cutoff cannot establish untouchability on its own.

Abstract Reasoning¶

Define \(s(n)=\sigma(n)-n\) on positive integers and fix the candidate value \(m\).
Search for structural preimage forms whose divisor sums can equal \(m\).
If a witness \(n\) is found, factor it and verify \(s(n)=m\) exactly.
If exclusion is sought, partition possible preimages by parity, factorization type, or divisor constraints.
Use identities such as \(s(p^2)=p+1\) to remove entire target families from the complement.
Distinguish an unconditional exclusion theorem from one depending on a Goldbach-type conjecture.
For density claims, count omitted image values with a stated asymptotic notion and error term.
For computation, justify the preimage bound and retain a reproducible certificate for each classified value.

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.

Examples¶

Canonical¶

The target \(4\) is touchable because \(9\) has positive divisors \(1,3,9\), hence \(\sigma(9)=13\) and \(s(9)=13-9=4\). The target \(6\) is touchable by the self-witness \(6\), since its proper divisors sum to \(1+2+3=6\). These examples demonstrate that classification belongs to the inverse image of \(s\), not to a superficial property of the target's own factorization.

Mapped back: target integer → search for aliquot-sum preimage → exact divisor certificate → touchable classification, with untouchable reserved for proved absence.

Applied / In Practice¶

To show that \(p+1\) is touchable for every prime \(p\), use the preimage \(p^2\). Its divisors are \(1,p,p^2\), so its proper divisors sum to \(1+p\). The argument covers an infinite family with one factorization pattern. It does not classify neighboring values, nor does it settle the odd-untouchable problem; those require other preimages or exclusion arguments.^[4]

Mapped back: structured target family → symbolic preimage construction → divisor identity → infinite exclusion from the untouchable complement.

Structural Tensions¶

Finite search vs. infinite exclusion. Missing a witness below a cutoff does not prove none exists. Diagnostic: What theorem bounds all possible preimages?
Forward iteration vs. inverse fibers. Aliquot sequences follow outputs while untouchability concerns incoming edges. Diagnostic: Is the argument moving in the correct direction?
Observed parity vs. proved parity. Known terms are mostly even, but the odd case is open. Diagnostic: Is a heuristic being reported as a theorem?
Lower density vs. natural density. A positive lower bound does not establish a limiting proportion. Diagnostic: Which density notion does the source prove?
Autonomous number class vs. generic Complement. Complements travel; the aliquot-sum image supplies this residual. Diagnostic: Are universe and mapping both fixed to positive integers and \(s\)?

Structural–Framed Character¶

The nonimage predicate is structural once the domain and aliquot-sum function are fixed. Naming, inclusion of zero in auxiliary codomains, computational cutoffs, and the choice of density statistic are framed conventions. The construct is domain-specific because its designated subset is generated by the arithmetic of proper divisors.

Structural Core vs. Domain Accent¶

The portable skeleton is declared universe minus designated subset. The domain accent is the positive integers, divisor-sum function, aliquot image, preimage certificates, parity problem, and nonaliquot density. Removing them leaves Complement; retaining them yields Untouchable Number.

Complement is the strict parent because the class is defined exactly as \(\mathbb N_{>0}\setminus s(\mathbb N_{>0})\). Natural Number supplies the carrier and Function Mapping supplies \(s\), but Complement expresses the defining relation most literally.

The prospective workspace queue contains one strict upward edge to prime:complement. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

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

Untouchable Number → Complement → Set and Membership

Neighborhood in Abstraction Space¶

Untouchable Number sits in a sparse region of the domain-specific corpus (87^th 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

Not to Be Confused With¶

Aliquot sum. The function value \(s(n)\), whose omitted outputs define the class.
Perfect number. A fixed point of \(s\), hence necessarily touchable.
Weird number. An abundant number lacking a subset-sum representation by proper divisors.
Nontotient. A positive integer outside the image of Euler's totient function.
Noncototient. A positive integer outside the image of \(n-\varphi(n)\).
Sociable number. A member of a cycle under repeated aliquot summation.

References¶

[1] Yong-Gao Chen and Qing-Qing Zhao, ‘Nonaliquot Numbers,’ Publicationes Mathematicae Debrecen 78, no. 2 (2011): 439–442, https://doi.org/10.5486/PMD.2011.4820. registry ↩a ↩b

[2] Jacques Sesiano, ‘Two Problems of Number Theory in Islamic Times,’ Archive for History of Exact Sciences 41, no. 3 (1991): 235–238, https://doi.org/10.1007/BF00348408. registry ↩

[3] Paul Erdős, ‘Über die Zahlen der Form \(\sigma(n)-n\) und \(n-\varphi(n)\),’ Elemente der Mathematik 28 (1973): 83–86, https://geodesic.mathdoc.fr/volume/ELMA_1973__28/. registry ↩

[4] Richard K. Guy, Unsolved Problems in Number Theory, 3^rd ed. (Springer, 2004), section B10, https://doi.org/10.1007/978-0-387-26677-0. registry ↩