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: v3 · 2026-09-19 · History
Domain-specific #: 1634
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Divisor Sum Classification → Mathematics

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.

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Not-Enough-Pieces Numbers

Take a number, like 8, and find all the smaller numbers that go into it evenly: 1, 2, and 4. Add them up: 1 + 2 + 4 = 7. Since 7 is less than 8, we call 8 a deficient number: its helpers add up to less than the number itself.

Numbers Whose Divisors Fall Short

A divisor of a number is a number that divides it evenly, with no remainder. The proper divisors are all the divisors except the number itself. A deficient number is a positive whole number whose proper divisors add up to less than the number. For example, the proper divisors of 10 are 1, 2, and 5, which add to 8, less than 10, so 10 is deficient. If they add to exactly the number, like 6 with 1 + 2 + 3 = 6, it is a perfect number instead, not a deficient one. And if they add to more, like 12, it is called abundant.

Deficient Numbers (Divisor Sum Below n)

A positive integer n is deficient when the sum of its proper divisors, s(n), is less than n. Since the sum of all divisors σ(n) equals s(n) + n, this is the same as σ(n) < 2n. The inequality is strict: when s(n) = n the number is perfect, and when s(n) > n it is abundant. Being deficient is not the same as being small, prime, or having few divisors, although primes and powers of primes are always deficient. To check, list or compute all positive divisors carefully, add them, subtract n if you used σ, and compare with a strict less-than.

A positive integer n is deficient when s(n) < n, where s(n) is the sum of its positive proper divisors; since s(n) = σ(n) − n, this is equivalent to σ(n) < 2n. The proper divisors encode n's multiplicative structure, and their sum gives a single abundance comparison, with the strict inequality placing n below the equality boundary occupied by perfect numbers. The concept is the exact divisor-sum inequality, not smallness, primality, or having few divisors. It breaks down if zero or negative integers are admitted without a new convention, if only some divisors are summed, if n is included in the sum while still comparing to n, or if equality is counted as deficiency. Checking it requires enumerating or factoring to get all positive divisors, summing them without omission or duplication, subtracting n when using σ, and applying the strict inequality. The classification supports partitioning integers by abundance, proving that classes such as primes and prime powers are deficient, and studying divisor-sum ratios.

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.
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) — outgoing typed relations

Deficient number is a kind of Constraint Prime

Membership is literally determined by the strict inequality constraint on an arithmetic function; What makes it its own entry: the exact proper-divisor-sum inequality on positive integers, rather than smallness, primality, scarcity of divisors, or an informal failure to reach a target.

Hierarchy path (1) — routes to 1 parentless root

Deficient number → is a kind of Constraint

Neighborhood in Abstraction Space¶

Deficient number sits in a crowded region of the domain-specific corpus (8^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Classifications & Arithmetic Functions (45 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25