Highly cototient number¶

A positive integer k>1 having more solutions to x−φ(x)=k than any smaller integer greater than one, where φ is Euler's totient function.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4880
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Totient Inverse Records → Mathematics

Core Idea¶

A highly cototient number is a record value of the multiplicity of the cototient map x↦x−phi(x), excluding k=1. Different factorizations of x can yield the same count of non-coprime residues; record holders are values where this inverse image is larger than for every prior eligible k. 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.

How would you explain it like I'm…

The Popular Answer Record

Imagine a number machine. You put in a number, and it tells you how many numbers from 1 up to yours share a building block with it. Lots of different starting numbers can give the same answer. A highly cototient number is an answer that more starting numbers lead to than any smaller answer (not counting 1). It's a record answer!

Most-Reached Cototient Answers

For any number x, count how many numbers from 1 to x share a common factor bigger than 1 with x. That count is called the cototient of x. Different numbers x can have the same cototient. For each answer k, count how many x's give it. A highly cototient number is an answer k, bigger than 1, that is reached by more x's than any smaller answer from 2 up to k−1. It sets a new record for 'most ways to get this answer'.

Inverse Cototient Records

Euler's phi function φ(x) counts the numbers from 1 to x that share no factor with x, so x − φ(x), the cototient, counts those that do share a factor. Many different x can have the same cototient because different factorizations can lead to the same count. For each k, look at how many positive integers x satisfy x − φ(x) = k. A highly cototient number is a k > 1 whose count of such x is strictly larger than the count for every integer from 2 up to k − 1. The value k = 1 is excluded from the comparison. It's a record condition on how many preimages a value has, not a property of any single x.

Define the cototient map c(x) = x − φ(x), which counts residues in [1, x] not coprime to x. For an integer k, let N(k) be the number of positive integers x with c(x) = k, the size of the inverse image of k. Because distinct factorizations of x can produce the same number of non-coprime residues, N(k) can exceed one. A highly cototient number is a k > 1 with N(k) strictly greater than N(j) for every j with 2 ≤ j ≤ k − 1. The case k = 1 is excluded from the record comparison. So the object of study is the record-multiplicity sequence for inverse cototient values, and confusing it with properties of the cototient itself, or of totient values, substitutes a neighboring concept.

Scope of Application¶

Highly cototient number belongs to number theory and is useful where the analyst can specify positive integers x and k, Euler's totient phi(x), the cototient x-phi(x), preimage counts and a record comparison over smaller k, then evaluate k exceeds one and its number of positive-integer preimages under x-phi(x) strictly exceeds the preimage count of every integer between two and k-1. The scope is broad within that domain but bounded by the need for k exceeds one and its number of positive-integer preimages under x-phi(x) strictly exceeds the preimage count of every integer between two and k-1. 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 k exceeds one and its number of positive-integer preimages under x-phi(x) strictly exceeds the preimage count of every integer between two and k-1 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 Highly cototient 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 Highly cototient number. Highly cototient 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: positive integers x and k, Euler's totient phi(x), the cototient x-phi(x), preimage counts and a record comparison over smaller k. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express k exceeds one and its number of positive-integer preimages under x-phi(x) strictly exceeds the preimage count of every integer between two and k-1 independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse positive integers x and k, Euler's totient phi(x), the cototient x-phi(x), preimage counts and a record comparison over smaller k, Different factorizations of x can yield the same count of non-coprime residues; record holders are values where this inverse image is larger than for every prior eligible k., and type the carrier, state every parameter and convention in the definition, test that k exceeds one and its number of positive-integer preimages under x-phi(x) strictly exceeds the preimage count of every integer between two and k-1, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Highly cototient number Domain-specific

Parents (1) — outgoing typed relations

Highly cototient number is a kind of Measurement Prime

The identity measures and records inverse-image multiplicity of an arithmetic function.

Hierarchy path (1) — routes to 1 parentless root

Highly cototient number → is a kind of Measurement

Neighborhood in Abstraction Space¶

Highly cototient number sits in a crowded region of the domain-specific corpus (18^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