Complete sequence¶

A sequence of natural numbers whose distinct finite subset sums represent every positive integer.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3795
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Additive Number Theory → Mathematics

Core Idea¶

A sequence is complete when each positive integer equals a sum of zero-or-one copies of its terms, with order, repetition, and positivity conventions explicitly fixed. Early terms create a covered interval; a new term no larger than one plus the previous covered sum extends that interval without gaps, supporting recursive completeness tests. 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…

Coins That Make Every Amount

Imagine a special set of coins, and you may use each coin only once. If you can make every amount, 1, 2, 3, and on and on, by picking some of those coins, then the list of coins is complete. Coins worth 1, 2, 4, 8 and so on work.

Every Number From Using Each Once

A complete sequence is a list of positive whole numbers where every positive whole number can be made by adding up some of them, using each one at most once. For example, 1, 2, 4, 8, 16 and so on is complete: 11 is 8 + 2 + 1. There is a handy check. If the numbers you already have can make every amount up to some total, then a new number that is no bigger than one more than that total keeps everything covered with no gaps. If a new number is too big, some amount gets skipped.

Sums of Distinct Terms Cover All Integers

A sequence of positive integers is complete if every positive integer can be written as a sum of some of its terms, using each term zero times or once. The conventions matter, such as whether terms can repeat in the list and whether they must be positive, so they should be stated. The standard test works step by step, with terms in increasing order. The first few terms can make every number from 1 up to their total sum. A new term that is at most one more than that sum extends the covered range with no gaps; a bigger term leaves a gap. The powers of two are the classic example of a complete sequence.

In additive number theory, a sequence of positive integers is complete if every positive integer has at least one finite representation as a sum of terms of the sequence, each term used at most once. Conventions about ordering, repetition, and positivity must be fixed explicitly. The key mechanism is the covered interval: if the initial terms can represent every integer from 1 up to their partial sum, then appending a term no larger than one plus that sum extends the covered interval without gaps. This gives a recursive test for completeness when the terms are sorted in nondecreasing order. The condition is existence of a representation, not uniqueness, so completeness should not be confused with unique-representation systems like binary.

Scope of Application¶

Complete sequence belongs to additive number theory and is useful where the analyst can specify the typed additive number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, then evaluate every positive integer has at least one finite representation using each sequence term at most once. The scope is broad within that domain but bounded by the need for every positive integer has at least one finite representation using each sequence term at most once. 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 every positive integer has at least one finite representation using each sequence term at most once 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 Complete sequence 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 Complete sequence. Complete sequence 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 additive 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.
Lock the constitutive rule. Express every positive integer has at least one finite representation using each sequence term at most once independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of additive number theory because they reuse the typed additive number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, Early terms create a covered interval; a new term no larger than one plus the previous covered sum extends that interval without gaps, supporting recursive completeness tests., and type the carrier, state every parameter and convention in the definition, test that every positive integer has at least one finite representation using each sequence term at most once, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Complete sequence Domain-specific

Parents (1) — outgoing typed relations

Complete sequence is a kind of Decomposition Prime

What makes it its own entry: the domain-specific identity determined by every positive integer has at least one finite representation using each sequence term at most once.

Hierarchy path (1) — routes to 1 parentless root

Complete sequence → is a kind of Decomposition

Neighborhood in Abstraction Space¶

Complete sequence 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 — Numeral Systems & Integer Sequences (22 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25