Hemiperfect number¶

A positive integer whose sum-of-divisors function divided by the integer is a half-integer with odd numerator.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4857
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

For n, the abundancy index sigma of n over n equals k over two for an odd integer k; the name is unrelated to semiperfect numbers, which concern subset sums of proper divisors. Prime factorization computes the multiplicative divisor-sum function, division by n gives the abundancy index and an odd-over-two reduction tests membership. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

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And-a-Half Numbers

Take a number and find all the numbers that fit into it evenly, including the number itself. Add them up. If the total is exactly one-and-a-half times the number, or two-and-a-half times, or any whole amount plus a half, it is a hemiperfect number. For 2, the numbers that fit are 1 and 2, and they add to 3, which is one-and-a-half times 2.

Half-Way Divisor Sums

For any whole number, list all the numbers that divide it evenly, including 1 and the number itself, and add them up. Then divide that sum by the original number. If the answer is an odd number divided by 2, such as 3/2 or 5/2, the number is hemiperfect. For example, 24's divisors add to 60, and 60 ÷ 24 = 5/2, so 24 is hemiperfect. Watch out: "semiperfect" is a completely different idea, even though the names sound similar.

Half-Integer Abundancy Numbers

Let σ(n) be the sum of all positive divisors of n. The ratio σ(n)/n is called the abundancy index. A positive integer n is hemiperfect when this ratio, written in lowest terms, equals k/2 with k odd. For example, σ(24) = 60 and 60/24 = 5/2. To test a number, factor it into primes, use the fact that σ is multiplicative to get σ(n), divide by n, and reduce the fraction. This is unrelated to semiperfect numbers, which are about whether some of a number's proper divisors add up to the number.

Hemiperfect numbers are defined through the abundancy index σ(n)/n, where σ is the sum of all positive divisors. A positive integer n is hemiperfect if σ(n)/n = k/2 in lowest terms for some odd integer k, so the index is a half-integer. Computation uses the prime factorization: σ is multiplicative, and σ(p^a) = (p^{a+1} − 1)/(p − 1). Dividing by n and reducing the fraction gives the membership test. For instance, n = 2 gives 3/2 and n = 24 gives 5/2. The conventions must be explicit: positive divisors, including n itself, and the ratio taken in lowest terms. Claims about the smallest hemiperfect number with a given k need evidence or bounds. The name is unrelated to semiperfect numbers, which concern subset sums of proper divisors.

Scope of Application¶

Hemiperfect number belongs to number theory and is useful where the analyst can specify the typed number theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the positive integer n, positive-divisor convention, sigma function, abundancy ratio in lowest terms, odd numerator k and evidence or bounds for any least-member claim are explicit. The scope is broad within that domain but bounded by the need for the positive integer n, positive-divisor convention, sigma function, abundancy ratio in lowest terms, odd numerator k and evidence or bounds for any least-member claim are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the positive integer n, positive-divisor convention, sigma function, abundancy ratio in lowest terms, odd numerator k and evidence or bounds for any least-member claim are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Hemiperfect number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Hemiperfect number. Hemiperfect number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed number theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the positive integer n, positive-divisor convention, sigma function, abundancy ratio in lowest terms, odd numerator k and evidence or bounds for any least-member claim are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse the typed number theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, Prime factorization computes the multiplicative divisor-sum function, division by n gives the abundancy index and an odd-over-two reduction tests membership., and type the carrier, state every parameter and convention in the definition, test that the positive integer n, positive-divisor convention, sigma function, abundancy ratio in lowest terms, odd numerator k and evidence or bounds for any least-member claim are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Hemiperfect number Domain-specific

Parents (1) — outgoing typed relations

Hemiperfect number is a kind of Ratio Prime

It remains its own entry because its identity is fixed by the positive integer n, positive-divisor convention, sigma function, abundancy ratio in lowest terms, odd numerator k and evidence or bounds for any least-member claim.

Hierarchy path (1) — routes to 1 parentless root

Hemiperfect number → is a kind of Ratio → is a kind of Comparison

Neighborhood in Abstraction Space¶

Hemiperfect number sits in a crowded region of the domain-specific corpus (4^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer & Divisor Number Theory (45 abstractions)

Nearest neighbors

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