Odious number¶

A nonnegative integer whose binary expansion contains an odd number of one bits.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5854
Origin domain: number theory
Subdomain: number theory

Core Idea¶

Odious numbers are the parity-one positions of the Thue–Morse sequence; integers with an even population count are conventionally called evil numbers. Binary representation decomposes the integer into powers of two, one bits are counted and parity of that count classifies the number. 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 the domain-specific identity determined by the nonnegative-integer domain, canonical binary representation without leading zeros, one-bit count and odd-parity condition are explicit.

Scope of Application¶

Odious number belongs to number theory and is useful where the analyst can specify the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the nonnegative-integer domain, canonical binary representation without leading zeros, one-bit count and odd-parity condition are explicit. The scope is broad within that domain but bounded by the need for the nonnegative-integer domain, canonical binary representation without leading zeros, one-bit count and odd-parity condition 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 nonnegative-integer domain, canonical binary representation without leading zeros, one-bit count and odd-parity condition 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 Odious 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 Odious number. Odious 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, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the nonnegative-integer domain, canonical binary representation without leading zeros, one-bit count and odd-parity condition 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, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Binary representation decomposes the integer into powers of two, one bits are counted and parity of that count classifies the number., and type the carrier, state every parameter and convention in the definition, test that the nonnegative-integer domain, canonical binary representation without leading zeros, one-bit count and odd-parity condition are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Odious number Domain-specific

Parents (1) — more general patterns this builds on

Odious number is a kind of Classification Prime

The proposed strict upward parent is prime:classification.

Hierarchy path (1) — routes to 1 parentless root

Odious number → Classification

Neighborhood in Abstraction Space¶

Odious number sits in a crowded region of the domain-specific corpus (5^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

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