Amenable number¶
A positive integer n admitting a multiset of exactly n integers whose sum and product are both n.
Core Idea¶
Negative and repeated integers are allowed, and the characterization selects numbers congruent to zero or one modulo four except four under the stated convention. Pairs of one and negative one preserve product while adjusting length and sum, and arithmetic constructions build a witness multiset for each admissible congruence class. 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.
Scope of Application¶
Amenable number belongs to recreational number theory and is useful where the analyst can specify the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit. The scope is broad within that domain but bounded by the need for the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction 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, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction 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 Amenable 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 Amenable number. Amenable 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 recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of recreational number theory because they reuse the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Pairs of one and negative one preserve product while adjusting length and sum, and arithmetic constructions build a witness multiset for each admissible congruence class., and type the carrier, state every parameter and convention in the definition, test that the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Amenable number Domain-specific
Parents (1) — more general patterns this builds on
-
Amenable number is a kind of Constraint Prime
The proposed strict upward parent is
prime:constraint.
Hierarchy path (1) — routes to 1 parentless root
- Amenable number → Constraint
Neighborhood in Abstraction Space¶
Amenable number sits in a crowded region of the domain-specific corpus (15th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Digit Properties & Recreational Numbers (6 abstractions)
Nearest neighbors
- Taxicab number — 0.93
- Multiplicative partition — 0.92
- 15 puzzle — 0.92
- Keith number — 0.92
- Arithmetic number — 0.91
Computed from structural-signature embeddings · 2026-09-08