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.
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 \(\sigma(n)<2n\). 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.
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.
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
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. 2. 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. 3.
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.
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.
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 (14th 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
- Quasiperfect number — 0.95
- Hyperperfect number — 0.95
- Sublime number — 0.93
- Perfect number — 0.93
- Multiply perfect number — 0.91
Computed from structural-signature embeddings · 2026-09-08