Deficient number¶

Classify a positive integer as deficient when the sum of its positive proper divisors is smaller than the integer itself, equivalently when its divisor sum is less than twice the integer.

Version: v2 · 2026-08-30 · History
Domain-specific #: 1634
Origin domain: number theory
Subdomain: divisor sum classification

Core Idea¶

A positive integer \(n\) is deficient when \(s(n)<n\), where \(s(n)\) is the sum of positive proper divisors; since \(s(n)=\sigma(n)-n\), this is equivalent to \(\sigma(n)<2n\).^[1] The proper divisors encode the integer's multiplicative substructure, their sum produces a scalar abundance comparison, and the strict inequality places the integer below the equality boundary occupied by perfect numbers.

Its autonomous residual is the exact proper-divisor-sum inequality on positive integers, rather than smallness, primality, scarcity of divisors, or an informal failure to reach a target. The identity fails when zero or a negative integer is admitted without a new convention, only selected divisors are summed, the integer itself is included while the wrong comparison is retained, or equality is misclassified as deficiency.

Recognition requires an analyst to factor the integer or enumerate its positive divisors, compute the divisor sum without omitting or duplicating factors, subtract the integer if using the full divisor-sum function, and compare using a strict inequality. Once established, it supports partitioning positive integers by divisor abundance, proving closure properties for classes of primes and prime powers, studying multiplicative divisor-sum ratios, and separating arithmetic classification from empirical pattern spotting without turning those uses into the definition.

Structural Signature¶

Carrier: a positive integer \(n\), its set of positive divisors, the proper-divisor sum \(s(n)\), and the divisor-sum function \(\sigma(n)\)
Inputs or antecedent state: the positive-integer domain, a divisor convention, exact factorization or divisor enumeration, summation, and a strict inequality
Constitutive operation: The proper divisors encode the integer's multiplicative substructure, their sum produces a scalar abundance comparison, and the strict inequality places the integer below the equality boundary occupied by perfect numbers
Invariant: the sum of all positive divisors smaller than \(n\) is strictly less than \(n\), under a fixed positive-divisor convention
Recognition test: factor the integer or enumerate its positive divisors, compute the divisor sum without omitting or duplicating factors, subtract the integer if using the full divisor-sum function, and compare using a strict inequality
Output or consequence: partitioning positive integers by divisor abundance, proving closure properties for classes of primes and prime powers, studying multiplicative divisor-sum ratios, and separating arithmetic classification from empirical pattern spotting
Failure boundary: zero or a negative integer is admitted without a new convention, only selected divisors are summed, the integer itself is included while the wrong comparison is retained, or equality is misclassified as deficiency

What It Is Not¶

It is not the whole field of number theory; many objects in that field do not satisfy its constitutive rule.
It is not its canonical example. Every prime number is deficient because its only positive proper divisor is 1. That is an instance, not a definition.
It is not Divisor Function. The divisor-sum function supplies the computed quantity; deficient number is the integer class selected by a particular strict inequality involving that function.
It is not an unrestricted metaphor. The integer 1 is deficient under the usual convention because its proper-divisor sum is the empty sum 0, while deficient perfect powers and odd deficient composites show that neither parity nor primality is constitutive

Scope of Application¶

Deficient number applies when the analyst can specify a positive integer \(n\), its set of positive divisors, the proper-divisor sum \(s(n)\), and the divisor-sum function \(\sigma(n)\) and establish that the sum of all positive divisors smaller than \(n\) is strictly less than \(n\), under a fixed positive-divisor convention. The standard identity is over positive integers. Generalizations to ideals, polynomials, or other divisor theories require separately defined sums and order relations.^[2]

Recognition. factor the integer or enumerate its positive divisors, compute the divisor sum without omitting or duplicating factors, subtract the integer if using the full divisor-sum function, and compare using a strict inequality
Comparison. Compare legitimate instances through integer size, prime factorization, divisor convention, proper-divisor sum, full divisor sum, abundancy ratio, parity, multiplicativity, and closure under multiplication.
Boundary. The integer 1 is deficient under the usual convention because its proper-divisor sum is the empty sum 0, while deficient perfect powers and odd deficient composites show that neither parity nor primality is constitutive
Use. Preserve every assumption when using the identity for partitioning positive integers by divisor abundance, proving closure properties for classes of primes and prime powers, studying multiplicative divisor-sum ratios, and separating arithmetic classification from empirical pattern spotting.

Clarity¶

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because deficient in ordinary language means inadequate, while the mathematical word records one exact strict inequality and makes no qualitative judgment. The disciplined statement is that the object counts as Deficient number exactly when the sum of all positive divisors smaller than \(n\) is strictly less than \(n\), under a fixed positive-divisor convention

Identity and measurement remain separate. Membership is proved by exact arithmetic; numerical approximations to a divisor ratio are not needed and can obscure the strict boundary at equality. Approximation or noisy evidence may weaken a classification without changing its definition.

Manages Complexity¶

The abstraction compresses primes, prime powers, odd and even composites, powers of two, multiplicatively constructed families, and comparisons through the abundancy index into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares integer size, prime factorization, divisor convention, proper-divisor sum, full divisor sum, abundancy ratio, parity, multiplicativity, and closure under multiplication and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning¶

Type the carrier. Establish a positive integer \(n\), its set of positive divisors, the proper-divisor sum \(s(n)\), and the divisor-sum function \(\sigma(n)\) and reject examples from a different problem.
Lock the rule. Express that the sum of all positive divisors smaller than \(n\) is strictly less than \(n\), under a fixed positive-divisor convention independently of one notation or implementation.
Derive carefully. Infer partitioning positive integers by divisor abundance, proving closure properties for classes of primes and prime powers, studying multiplicative divisor-sum ratios, and separating arithmetic classification from empirical pattern spotting only under the stated assumptions.
Stress-test. Contrast the legitimate boundary case—The integer 1 is deficient under the usual convention because its proper-divisor sum is the empty sum 0, while deficient perfect powers and odd deficient composites show that neither parity nor primality is constitutive—with this counterexample: 6 is not deficient because its proper divisors 1, 2, and 3 sum to 6, making it perfect rather than deficient.

Knowledge Transfer¶

Transfer within number theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from Every prime number is deficient because its only positive proper divisor is 1. to The integer 8 is deficient even though it is composite. demonstrates that continuity.^[3]

Outside the domain, only the skeleton—compute a summary over proper parts and classify the whole by whether that summary falls below it—travels automatically. The terms positive integer, divisor, proper divisor, aliquot sum, divisor-sum function, perfect, abundant, deficient, and abundancy index retain domain-specific meanings, so every role and inference must be revalidated.

Examples¶

Canonical¶

Every prime number is deficient because its only positive proper divisor is 1. For a prime \(p>1\), \(s(p)=1<p\); this proves deficiency directly without suggesting that all deficient numbers are prime. It is canonical because the carrier, rule, invariant, and consequence are all inspectable.^[1]

Mapped back: a positive integer \(n\), its set of positive divisors, the proper-divisor sum \(s(n)\), and the divisor-sum function \(\sigma(n)\) → The proper divisors encode the integer's multiplicative substructure, their sum produces a scalar abundance comparison, and the strict inequality places the integer below the equality boundary occupied by perfect numbers → the sum of all positive divisors smaller than \(n\) is strictly less than \(n\), under a fixed positive-divisor convention → partitioning positive integers by divisor abundance, proving closure properties for classes of primes and prime powers, studying multiplicative divisor-sum ratios, and separating arithmetic classification from empirical pattern spotting

Applied / In Practice¶

The integer 8 is deficient even though it is composite. Its positive proper divisors are \(1,2,4\), whose sum is \(7<8\), showing that compositeness and deficiency are logically independent classifications. It qualifies only after the same diagnostic and failure boundary are checked.^[2]

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Exact identity vs. practical recognition. The constitutive condition may be exact while evidence is indirect. Diagnostic: Can the reviewer state both the condition and the warrant?
T2: Canonical form vs. variants. primes, prime powers, odd and even composites, powers of two, multiplicatively constructed families, and comparisons through the abundancy index can preserve or change the identity. Diagnostic: Which named role is invariant across the variants?
T3: Compression vs. hidden assumptions. The label is useful only while prerequisites remain visible. Diagnostic: Can each downstream inference be traced to a declared assumption?
T4: Autonomy vs. reduction. The candidate uses broader structures but claims the exact proper-divisor-sum inequality on positive integers, rather than smallness, primality, scarcity of divisors, or an informal failure to reach a target. Diagnostic: Does that residual still support independent recognition after the parent and neighbors are subtracted?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is compute a summary over proper parts and classify the whole by whether that summary falls below it; its identity-bearing terms are positive integer, divisor, proper divisor, aliquot sum, divisor-sum function, perfect, abundant, deficient, and abundancy index. Those terms determine admissible objects, evidence, and consequences inside number theory.

Structural Core vs. Domain Accent¶

The structural core is a carrier governed by The proper divisors encode the integer's multiplicative substructure, their sum produces a scalar abundance comparison, and the strict inequality places the integer below the equality boundary occupied by perfect numbers and tested by factor the integer or enumerate its positive divisors, compute the divisor sum without omitting or duplicating factors, subtract the integer if using the full divisor-sum function, and compare using a strict inequality. The domain accent is constitutive rather than decorative, so an analogy that preserves only the skeleton is not another instance of Deficient number.

The proposed strict upward parent is prime:constraint. Membership is literally determined by the strict inequality constraint on an arithmetic function; integer divisors and their sum supply the domain-specific residual. The edge is proposal-only and points to a frozen prior-baseline Prime.

The entry does not collapse into the parent because the exact proper-divisor-sum inequality on positive integers, rather than smallness, primality, scarcity of divisors, or an informal failure to reach a target A thematic neighbor is declined whenever it does not literally subsume that rule.

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

Relationships to Other Abstractions¶

Current abstraction Deficient number Domain-specific

Parents (1) — more general patterns this builds on

Deficient number is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Deficient number → Constraint

Neighborhood in Abstraction Space¶

Deficient number sits in a crowded region of the domain-specific corpus (14^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Functions & Special Numbers (13 abstractions)

Nearest neighbors

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

Not to Be Confused With¶

Perfect number. Has proper-divisor sum equal to the integer.
Abundant number. Has proper-divisor sum greater than the integer.
Prime number. Always deficient beyond 1 but does not exhaust the deficient class.
Untouchable number. Cannot be expressed as a proper-divisor sum of another positive integer, a different relational property.

References¶

[1] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6^th ed., Oxford University Press, 2008, chapters on arithmetic functions and perfect numbers, ISBN 978-0-19-921986-5. registry ↩a ↩b

[2] Richard K. Guy, Unsolved Problems in Number Theory, 3^rd ed., Springer, 2004, sections B on divisor and aliquot problems, DOI 10.1007/978-0-387-26677-0. registry ↩a ↩b

[3] József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, chapters on divisor functions, DOI 10.1007/1-4020-2547-5. registry ↩