Ulam number¶
Generate a seed-dependent increasing integer sequence by repeatedly choosing the least larger integer having exactly one representation as a sum of two distinct earlier terms.
Core Idea¶
For seeds u<v, an Ulam sequence repeatedly adjoins the least integer larger than the current maximum that has exactly one representation as a sum of two distinct earlier sequence terms. A greedy recurrence recomputes representation multiplicities among earlier distinct terms and selects the least uniquely representable candidate. 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 additive number theory and integer sequences. It is the conjunction of greedy minimality, distinct-summand use, exact uniqueness, and dependence on the evolving admitted set.
Scope of Application¶
Ulam number belongs to additive number theory and integer sequences and is useful where the analyst can specify an increasing sequence of integers beginning from declared seeds, then evaluate admission is controlled by exact one-representation status relative to the whole earlier prefix. The scope is broad within that domain but bounded by the need for each post-seed term is the least larger integer with representation count exactly one among unordered pairs of distinct earlier terms. Computational patterns and empirical densities should not be promoted to theorems without proof.
Clarity¶
The abstraction clarifies a crowded vocabulary by making admission is controlled by exact one-representation status relative to the whole earlier prefix 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 Ulam's name also occurs in the unrelated Ulam spiral and Ulam–Harris tree.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: quadratically many pair sums in a naive implementation, dynamically changing uniqueness, long gaps, seed dependence, and sparse asymptotic evidence. Ulam 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: an increasing sequence of integers beginning from declared seeds. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express each post-seed term is the least larger integer with representation count exactly one among unordered pairs of distinct earlier terms independently of one notation or implementation. This step prevents the canonical example from becoming the definition.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of additive number theory and integer sequences because they reuse an increasing sequence of integers beginning from declared seeds, A greedy recurrence recomputes representation multiplicities among earlier distinct terms and selects the least uniquely representable candidate., and enumerate pair sums from the admitted prefix, count distinct representations, and verify minimality above the last term. A theorem, diagnostic, or modeling warning can travel when those roles remain literal.
Relationships to Other Abstractions¶
Current abstraction Ulam number Domain-specific
Parents (1) — more general patterns this builds on
-
Ulam number is a kind of Recurrence Prime
The proposed strict upward parent is
prime:recurrence.
Hierarchy path (1) — routes to 1 parentless root
- Ulam number → Recurrence
Neighborhood in Abstraction Space¶
Ulam number sits in a moderately populated region (53rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Recursive Patterns & Decomposition (5 abstractions)
Nearest neighbors
- Complete sequence — 0.89
- Random seed — 0.88
- Hyperharmonic number — 0.88
- Piecewise syndetic set — 0.88
- Fermat number — 0.88
Computed from structural-signature embeddings · 2026-09-08