Charting the landscape of mathematics through 394,307 sequences
The Online Encyclopedia of Integer Sequences contains 394,307 sequences submitted by mathematicians over decades. Each sequence comes with a name describing what it counts, measures, or represents. Together, these names form a kind of census of mathematics — a record of what mathematicians have found interesting enough to catalogue.
What happens when you map the connections between these descriptions? When "prime" appears alongside "partition" in dozens of sequences, that's a signal of a genuine mathematical relationship. When "graph" and "polynomial" co-occur, it points to chromatic polynomials, Tutte polynomials, and a web of connections between combinatorics and algebra. The names of integer sequences encode the structure of mathematics itself.
I downloaded all 394,307 sequence names, extracted mathematical concepts from each, measured which concepts co-occur more often than chance would predict, and built a network. The result is a map of mathematical concepts as reflected by the OEIS — not a map of what mathematics is, but of what mathematicians have studied.
Each node is a mathematical concept. Size reflects how many OEIS sequences mention it. Color indicates community membership (detected automatically by clustering algorithm). Edges connect concepts that co-occur in sequence descriptions more often than chance predicts. Hover over nodes for details. Click to highlight connections. Scroll to zoom. Drag to pan.
Each of the 394,307 OEIS sequence names was parsed to extract mathematical keywords. Compound terms like "continued fraction", "cellular automaton", and "Dyck path" were identified as single concepts. Common English words and generic mathematical terms ("number", "sequence", "integer") were filtered out, leaving concepts that carry genuine mathematical meaning.
Two concepts are "connected" if they appear together in sequence descriptions more often than chance would predict. The strength of each connection is measured using pointwise mutual information (PMI) — a standard information-theoretic measure of association:
If "prime" and "fibonacci" appear together in 2,000 sequences, and each appears in roughly 30,000
sequences out of 394,307 total, the PMI tells us whether 2,000 co-occurrences is more or less than
we'd expect from independent keywords. The edge weight combines PMI with absolute count:
score = PMI × √count, balancing statistical surprise with empirical frequency.
The Louvain algorithm partitions the network into communities by maximizing modularity — finding groups of concepts more densely connected to each other than to the rest of the network. The algorithm discovered 19 communities, each corresponding roughly to an area of mathematics.
Each community represents a cluster of tightly connected mathematical concepts. The labels below are derived from the most prominent concept in each cluster.
| Community | Size | Key Concepts |
|---|
The most interesting features of any map are the connections between regions. These "bridge" concepts appear in multiple mathematical contexts, linking areas that might seem unrelated. A concept that connects the primes cluster to the partitions cluster is revealing a genuine mathematical relationship — and the OEIS has documented it through hundreds of sequences.
Textbooks organize mathematics into a hierarchy: algebra, analysis, geometry, combinatorics. But the OEIS concept map shows a densely connected network where every area touches every other area. "Prime" connects to "partition" connects to "graph" connects to "polynomial" connects back to "prime." Mathematics isn't a tree — it's a web.
Some concepts appear everywhere. "Prime" is the most connected single concept, appearing in over 42,000 sequences and linking to nearly every community. This isn't surprising — primes are the atoms of number theory, and number theory touches everything. But the second tier of hubs is more revealing: "binary", "partition", "triangle", "square" — these are the connective tissue of mathematics, the concepts general enough to appear across domains but specific enough to carry meaning.
Cross-community edges reveal unexpected connections. When "graph" and "algebra" concepts co-occur, it reflects a deep mathematical reality: algebraic graph theory, spectral methods, and group-theoretic approaches to combinatorics. When "cellular automaton" concepts connect to "representation" concepts, it suggests that rule-based discrete dynamics and algebraic structure share more common ground than a textbook table of contents would suggest.
This map reflects the OEIS, not mathematics itself. The OEIS skews heavily toward discrete mathematics, combinatorics, and number theory. Continuous mathematics (analysis, differential equations, topology) is underrepresented because its objects — functions, manifolds, measure spaces — don't naturally produce integer sequences. The map is a portrait of mathematics as seen through a very particular lens: the lens of integers.
Data: All 394,307 sequences from oeis.org/stripped.gz and oeis.org/names.gz, downloaded March 22, 2026.
Concept extraction: Regular expression tokenization with compound-term detection (53 bigram patterns), stop-word removal, and frequency filtering (df ≥ 30). 2,682 candidate concepts, top 400 by total PMI score selected for visualization.
Network construction: PMI-weighted edges, minimum co-occurrence count of 10, KNN pruning (k=6) to control density. Community detection via Louvain algorithm (resolution=1.0). Layout computed by force-directed spring embedding (200 iterations).
Centrality metrics: PageRank (for identifying important concepts) and betweenness centrality (for identifying bridge concepts connecting different communities).