Sublime number¶

A positive integer having a perfect number of positive divisors and a divisor sum that is itself a perfect number.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6974
Origin domain: number theory
Subdomain: divisor function sequences

Core Idea¶

A sublime number is an integer n for which both tau(n) and sigma(n) are perfect numbers. Prime factorization determines the multiplicative divisor-count and divisor-sum functions, whose resulting values are tested against the perfect-number condition. 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.

The load-bearing residual is not the broad topic of number theory. It is simultaneous perfectness of two divisor-function values. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that all positive divisors are counted, their sum includes n, and both derived integers satisfy the standard perfect-number definition fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application¶

Sublime number belongs to number theory and is useful where the analyst can specify a positive integer n, divisor-count function tau(n), divisor-sum function sigma(n), perfect-number predicate, factorization and known or candidate sequence terms, then evaluate all positive divisors are counted, their sum includes n, and both derived integers satisfy the standard perfect-number definition. The scope is broad within that domain but bounded by the need for all positive divisors are counted, their sum includes n, and both derived integers satisfy the standard perfect-number definition. 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 all positive divisors are counted, their sum includes n, and both derived integers satisfy the standard perfect-number definition 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 Sublime 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 Sublime number. Sublime 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: a positive integer n, divisor-count function tau(n), divisor-sum function sigma(n), perfect-number predicate, factorization and known or candidate sequence terms. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express all positive divisors are counted, their sum includes n, and both derived integers satisfy the standard perfect-number definition independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse a positive integer n, divisor-count function tau(n), divisor-sum function sigma(n), perfect-number predicate, factorization and known or candidate sequence terms, Prime factorization determines the multiplicative divisor-count and divisor-sum functions, whose resulting values are tested against the perfect-number condition., and type the carrier, state every parameter and convention in the definition, test that all positive divisors are counted, their sum includes n, and both derived integers satisfy the standard perfect-number definition, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Sublime number Domain-specific

Parents (1) — more general patterns this builds on

Sublime number is a kind of Classification Prime

The proposed strict upward parent is prime:classification.

Hierarchy path (1) — routes to 1 parentless root

Sublime number → Classification

Neighborhood in Abstraction Space¶

Sublime 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 Functions & Special Numbers (13 abstractions)

Nearest neighbors

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