Sum-Free Sequence¶
A strictly increasing sequence of positive integers in which no term is representable as a sum of a subset of its predecessors, coupling prefix-dependent additive avoidance to sparse-growth and reciprocal-sum questions.
Core Idea¶
A sum-free sequence in this number-theoretic sense is a strictly increasing sequence of positive integers
such that for every \(n\), the term \(a_n\) is not a sum of any subset of its predecessors:
Because the terms are distinct, this is equivalently avoidance of sums of distinct smaller sequence elements. Deshouillers, Erdős, and Melfi use this exact prefix/subset-sum identity and study how slowly such a sequence can grow.
Scope of Application¶
The abstraction belongs to additive number theory and extremal integer-sequence research. Questions include reciprocal-sum maxima, counting-function bounds, possible polynomial growth, and gaps \(a_{n+1}-a_n\). Deshouillers, Erdős, and Melfi prove the existence of polynomial-growth examples with successive-term ratio tending to one, showing that sum-free sequences need not look exponentially spaced.
Levine's extremal result studies sequences in which no term is a sum of distinct other terms and proves a sharp asymptotic reciprocal-sum bound when the first term is constrained. These results demonstrate a stable literature identity beyond a single puzzle.
Clarity¶
The powers of two
form a sum-free sequence because
is one larger than the sum of all preceding powers. Therefore no subset of predecessors can reach \(2^n\).
By contrast, \(1,3,4,10,\ldots\) fails at the third term because \(4=1+3\). The pair \(1+3\) is a predecessor subset.
Manages Complexity¶
Naively, testing \(a_n\) appears to require enumerating \(2^{n-1}\) predecessor subsets. The abstraction separates identity from algorithms: dynamic subset-sum methods, generating functions, modular arguments, or sufficient growth conditions can certify avoidance without listing every subset.
At the research level, the definition compresses a tension between additive richness and sequence density. A dense collection has many subset sums and is more likely to represent a later element; a sparse collection avoids collisions but yields small reciprocal mass.
Abstract Reasoning¶
For a finite prefix define its subset-sum set
Then legality of the next term is exactly \(a_n\notin\Sigma_{n-1}\), together with \(a_n>a_{n-1}\). When \(a_n\) is added, the new subset-sum set is
Knowledge Transfer¶
The portable pattern is prefix-generated forbidden set + choose the next element outside it. Similar online constructions occur in coding, combinatorial designs, and conflict-free scheduling. Literal transfer requires a commutative sum operation, one-use predecessor selection, strict order, and avoidance of the generated subset-sum set.
The candidate does not rise to prime status because its mathematical operation and integer-order assumptions are constitutive. Generic Constraint and Sequencing capture parts of the skeleton, while the additive subset-sum closure is the domain-specific residue.
Relationships to Other Abstractions¶
Current abstraction Sum-Free Sequence Domain-specific
Parents (1) — more general patterns this builds on
-
Sum-Free Sequence presupposes Constraint Prime
Sum-Free Sequence compositionally presupposes Constraint: at each index, the next integer must lie outside the subset-sum closure of the predecessor prefix.
Hierarchy path (1) — routes to 1 parentless root
- Sum-Free Sequence → Constraint
Neighborhood in Abstraction Space¶
Sum-Free Sequence sits in a sparse region of the domain-specific corpus (82nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Additive Number Theory & Series Tests (6 abstractions)
Nearest neighbors
- Primefree Sequence — 0.83
- Schnirelmann Density — 0.83
- Refactorable number — 0.81
- Ulam number — 0.81
- Pascal's rule — 0.81
Computed from structural-signature embeddings · 2026-09-08