Friendly number¶

A positive integer sharing its abundancy index—the sum of divisors divided by the integer—with at least one distinct positive integer.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4623
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

Friendly is relational rather than a claim of abundance, solitary status may be unknown and the divisor-sum convention includes all positive divisors. The sum-of-divisors function is normalized by n, and integers with equal rational values are grouped into equivalence classes; members of non-singleton classes are friendly. 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 fixed by the positive integer n, divisor-sum function sigma, abundancy index sigma(n)/n, a distinct integer m with equal index, equivalence relation and friendly class, pair or tuple terminology, solitary numbers and known versus unresolved status are explicit.

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Same-Score Number Buddies

Every whole number can be given a special score by adding up all the numbers that divide it evenly, including itself, and then dividing by the number. Two different numbers are 'friends' if they get the very same score. A number is friendly if it has at least one friend like that. For some numbers, nobody knows yet whether a friend exists.

Matching Divisor Scores

Take a number and add up all of its divisors, including 1 and the number itself. Then divide that sum by the number to get its 'abundancy' score. For example, 6 has divisors 1, 2, 3 and 6, which add to 12, and 12 divided by 6 is 2. The number 28 also gets a score of 2, so 6 and 28 are friendly numbers. Being friendly is about sharing a score with another number, not about having a big score, and some numbers are 'solitary' with no friend, while for others mathematicians still don't know.

Equal Abundancy Index Classes

For a positive integer n, let σ(n) be the sum of all its positive divisors, including 1 and n. The abundancy index is σ(n)/n. Two distinct integers are friendly if they have the same abundancy index, and numbers with equal index form a group called a friendly class. Any number in a class with at least two members is friendly; a number whose index no other number shares is solitary. Friendliness is a relationship between numbers, not a claim that a number is abundant. For many numbers it is still unknown whether they are friendly or solitary.

Let σ(n) denote the sum of all positive divisors of n, and define the abundancy index I(n) = σ(n)/n, a rational number. The relation 'I(m) = I(n)' is an equivalence relation on positive integers, partitioning them into friendly classes. An integer is friendly if its class contains some distinct m, and solitary if its class is a singleton. Two members are called a friendly pair, and larger groups friendly tuples. The concept is relational: it concerns sharing an index, not having a large one, so it differs from abundance or perfection. Status can be unresolved, since for many integers it is not known whether a friend exists. Correct use requires fixing the divisor-sum convention (all positive divisors) and distinguishing proven solitary or friendly status from open cases.

Scope of Application¶

Friendly number belongs to number theory and is useful where the analyst can specify the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the positive integer n, divisor-sum function sigma, abundancy index sigma(n)/n, a distinct integer m with equal index, equivalence relation and friendly class, pair or tuple terminology, solitary numbers and known versus unresolved status are explicit. The scope is broad within that domain but bounded by the need for the positive integer n, divisor-sum function sigma, abundancy index sigma(n)/n, a distinct integer m with equal index, equivalence relation and friendly class, pair or tuple terminology, solitary numbers and known versus unresolved status are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the positive integer n, divisor-sum function sigma, abundancy index sigma(n)/n, a distinct integer m with equal index, equivalence relation and friendly class, pair or tuple terminology, solitary numbers and known versus unresolved status 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.

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 Friendly number. Friendly 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 objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the positive integer n, divisor-sum function sigma, abundancy index sigma(n)/n, a distinct integer m with equal index, equivalence relation and friendly class, pair or tuple terminology, solitary numbers and known versus unresolved status 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 objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, The sum-of-divisors function is normalized by n, and integers with equal rational values are grouped into equivalence classes; members of non-singleton classes are friendly., and type the carrier, state every parameter and convention in the definition, test that the positive integer n, divisor-sum function sigma, abundancy index sigma(n)/n, a distinct integer m with equal index, equivalence relation and friendly class, pair or tuple terminology, solitary numbers and known versus unresolved status are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Friendly number Domain-specific

Parents (1) — outgoing typed relations

Friendly number is a kind of Classification Prime

It remains its own entry because its identity is fixed by the positive integer n, divisor-sum function sigma, abundancy index sigma(n)/n, a distinct integer m with equal index, equivalence relation and friendly class, pair or tuple terminology, solitary numbers and known versus unresolved status.

Hierarchy path (1) — routes to 1 parentless root

Friendly number → is a kind of Classification

Neighborhood in Abstraction Space¶

Friendly number sits in a crowded region of the domain-specific corpus (2^nd 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