decompwlj 3D
An interactive atlas of integer sequences, seen through one simple decomposition.
5000 sequences from the OEIS, 10⁵ terms each, as a gallery of plates and an
explorable 3-D point cloud.
▶ Live site: decompwlj.net · How it works ·
All sequences ·
The idea in one line
Every term of a strictly increasing sequence is written as a product plus a jump:
a(n) = k · L + d jump d = a(n+1) − a(n) · weight k = the least divisor of a − d above d · level L = (a − d)/k
For example, the prime 113 is followed by 127, so d = 14 and a − d = 99. The least divisor of 99
above 14 is 33, so 113 = 33 × 3 + 14. A term is in the level class when k > L (orange) and in
the weight class when k ≤ L (blue). It is a Euclidean division of a by k whose remainder is the
jump; for the natural numbers the weight is the smallest prime factor of a − 1, the sieve of
Eratosthenes. The How it works page explains it step by step with a
live example. Background: decompwlj.com and
arXiv:0711.0865.
A quick tour
| |
|---|
| |
| Gallery — 5000 plates, search by name, A-number or family, random sequence. | k–L view — a card opens the flat plate; drag it and it tilts smoothly into 3-D. Here the natural numbers: the sieve of Eratosthenes. |
| |
| Compare two sequences, overlaid or side by side, with one camera. | A page per sequence — counts, notes, OEIS link and a CSV of all 10⁵ terms. |
Features
- 🖼️ Gallery of 5000 weight–level plates with search, a family selector, a random button,
sorting (A-number, name, level share, ties, forced level, largest terms) and filters
(the OEIS keywords core and nice, with ties, every term decomposes, level ≥ 50 %, forced
level ≥ 90 %), remembered in the browser;
cards and images load as you scroll. A sequence of the day (from the OEIS core and nice
ones) and the recently viewed sequences (kept in the browser) sit at the top; on phones the
intro shortens and the sort and filters fold behind one Filters button.
- 🧊 3-D viewer (three.js / WebGL) of log k, log L, log d: preset views (3D, k–L, k–d, L–d,
edge), point size and opacity, density mode, the L = 1 line, a filter on n and a sweep along it.
Changing view is one smooth camera flight (the perspective flattens into the 2-D views and back,
a "dolly zoom"), and a new sequence fades in. The views fit the free space of any screen, from a
phone (portrait or sideways) to QHD and UHD, with points and labels scaled to the canvas.
- 🧭 Similar sequences: the eight nearest in the catalogue's decomposition figures (level, tie,
L = 1, forced and decomposable shares; ranges of a, k, L and d), each one click from the viewer
or from a comparison.
- ⚖️ Compare two sequences, overlaid or side by side.
- 🔗 Shareable links — the URL records the sequence, view, mode and comparison.
- 📄 Static page per sequence for search engines, with a CSV download (
n;a;weight;level;jump,
rebuilt in the browser) and its own link-preview image.
- 🎓 How it works page with theory and a live example.
- 📱 On phones the controls open as a bottom sheet with the plot kept in view above it (Done,
or swipe the handle down, to close), the family chips scroll sideways, and buttons are sized for
fingers.
- 🎬 Video export: the video button makes a 6 s MP4 of the current sequence: 1 s on the
flat plate on screen (k–L, k–d, L–d or edge), then 5 s of one continuous move into 3-D, the
camera rising to look down on the cloud, with title, labels and legend. Each frame is rendered at its exact
time and encoded by the browser (WebCodecs H.264, packed with the vendored mp4-muxer), so the
film is smooth on any machine; browsers without it record live (WebM).
- 🌗 Light and dark themes, keyboard shortcuts, PNG snapshots.
- 📦 Plain static files: no server code, no database, no build step.
- 🔒 Privacy-friendly visit counts (): no cookies, no personal data.
Run it locally
git clone https://github.com/decompwlj/decompwlj3dopus.git
cd decompwlj3dopus
python3 -m http.server 8000 # then open http://localhost:8000/
Any static server works (npx serve, php -S localhost:8000, …). Opening index.html straight
from disk does not: browsers block fetch() on file://, and the page says so.
Visitors need a current browser with WebGL, import maps and DecompressionStream
(Chrome/Edge 111+, Firefox 113+, Safari 16.4+). The gallery works without WebGL.
Deploy
The site is the repository root. All paths are relative, so it works at a domain root or in a
subfolder.
On GitHub Pages
- Push the repository to GitHub (or fork it).
- Settings → Pages → Build and deployment: Source Deploy from a branch, branch main,
folder / (root), Save.
- A minute or two later the site is at
https://<user>.github.io/<repository>/
(the Actions tab shows each publication). Every push to main republishes it.
The empty .nojekyll file makes Pages serve the files as they are. The site is about 945 MB
against Pages' 1 GB limit (soft bandwidth limit: 100 GB a month), which leaves room for about
250 more sequences; beyond that the data would have to live on another host.
With a custom domain and HTTPS
This repository serves decompwlj.net through its CNAME file. For your own domain, put its
name in CNAME (or set it in Settings → Pages → Custom domain), then add the DNS records
below at your registrar. When the DNS check passes, GitHub issues a Let's Encrypt certificate
(minutes, sometimes up to 24 h); then tick Enforce HTTPS.
| Type | Name | Value |
|---|
| A | @ | 185.199.108.153, 185.199.109.153, 185.199.110.153, 185.199.111.153 (four records) |
| AAAA | @ | 2606:50c0:8000::153, 2606:50c0:8001::153, 2606:50c0:8002::153, 2606:50c0:8003::153 (four records) |
| CNAME | www | <user>.github.io. |
- Remove any other A/AAAA record on the apex, and any registrar "web redirection" or parking
page: they block HTTPS.
- If the zone has a CAA record, it must allow
letsencrypt.org.
- Recommended: verify the domain under your account's Settings → Pages (a TXT record), so no
one else can use it on GitHub Pages.
On any other static host
Upload the repository root (at least index.html, data/, thumbs/, vendor/, plus seq/,
family/, learn/, share/ for the static pages) to Apache, nginx, Caddy, Netlify, Cloudflare
Pages, S3 + CDN… The server must:
- serve
data/seq/**/chunk-NNN.bin.gz either as a raw gzip file or with Content-Encoding: gzip
(the page handles both);
- return 404 for missing files, not
index.html;
- not put authentication in front of the data.
Ready-made configurations are in deploy/ (apache-vhost.conf, nginx.conf).
Long cache lifetimes for data/, thumbs/, share/ and vendor/ are safe.
Search engines and link previews
tools/seo.py writes the static pages that search engines can index (the app itself lives in a
URL fragment): one page per sequence, the full list, one page per family, the How it works page,
404.html (which redirects /A000040 to /seq/A000040/), sitemap.xml, robots.txt, og.png
and a 600 × 315 share image per sequence. The base URL comes from CNAME; without one, pass
python3 tools/seo.py --base https://<user>.github.io/<repo>.
To get indexed: add the domain to Google Search Console (verify with a DNS TXT record) and
submit https://<domain>/sitemap.xml; do the same in Bing Webmaster Tools (it also feeds
DuckDuckGo and Yahoo). On GitHub, fill the repository's About box (description, website,
topics) and upload og.png under Settings → General → Social preview.
Visit counts (GoatCounter)
Every page loads GoatCounter, a free, open-source counter for
non-commercial sites: no cookies, no personal data, nothing stored in the browser, so no
consent banner is needed. The gallery counts as /, each sequence opened in the viewer as
/#A000040 (whatever its view), the static pages under their own path, and each CSV download as
the event csv/A000040. Visits from localhost are ignored, and the site works the same if a
blocker stops the counter.
To turn it on for decompwlj.net:
- Sign up at goatcounter.com/signup with the code
decompwlj: the dashboard is then https://decompwlj.goatcounter.com.
- In its Settings, set the site domain to
decompwlj.net; optionally make the dashboard
public, or ignore your own visits from the dashboard's settings.
Counts appear as soon as the pages are live. To use another code, change the URL in the
data-goatcounter script of index.html and GOATCOUNTER_URL in tools/seo.py, then run
python3 tools/seo.py; to remove the counter, delete that script and set GOATCOUNTER_URL = ''.
Updating
Merge into main (GitHub Pages) or replace the files. If a visitor still sees the old version,
a hard reload (Ctrl+F5, Cmd+Shift+R) clears the cached index.html and data/catalog.csv.
Using the site
| Link | Opens |
|---|
/ or /#home | the gallery |
/#A000040 or /#primes | a sequence in the viewer |
/#primes.kLd.iso.solid.one | …with a view (iso, xy, xz, yz, edge), a point mode (solid, density) and the L = 1 highlight |
/#primes.kLd.iso.solid.vs-A001359 | a comparison (add .split for side by side) |
/seq/A000040/ (or /A000040) | the sequence's static page, with Download CSV |
/seq/, /family/primes/, /learn/ | all sequences, one family, how it works |
Keys — gallery: / search, R random, T theme, Home top. Viewer: G/Esc gallery,
↑/↓ neighbouring sequence, R random, 1–5 views, V overlay/side by side, C/P side
panels, T theme, Space sweep along n.
Opened from a gallery card or from its sequence page (click the plate), a sequence starts in the
flat k–L view, the same picture as the plate; the first grab-and-drag tilts it into 3-D. A link
that names a view keeps it. In the viewer, sequence page (next to the A-number, and in the
Sequence panel) leads back to the sequence's page; browsers that support it cross-fade between the two.
Under the hood
| Path | Contents |
|---|
index.html | The whole app: gallery, viewer, CSS and JavaScript (~110 kB) |
data/catalog.csv | One row per sequence: id, A-number, OEIS name, family, index range, counts, ranges, note |
data/seq/<id>/chunk-NNN.bin.gz | The data: 50,000 terms per chunk, compact binary, gzip |
thumbs/<id>.webp | Gallery plates (480 × 480, transparent, lossless WebP) |
share/<id>.jpg | Link previews of the sequence pages (600 × 315) |
seq/, family/, learn/, 404.html, sitemap.xml, robots.txt, og.png | Static pages, written by tools/seo.py |
css/pages.css | Stylesheet of the static pages |
vendor/ | three.js r169 and OrbitControls, unmodified; mp4-muxer 5.2.2, minified (MIT) |
tools/ | Generator, OEIS metadata, encoders, audit and page builders (not needed at runtime) |
tests/smoke.mjs, .github/workflows/checks.yml | Automatic checks |
deploy/ | Apache and nginx examples |
docs/SEQUENCES.md | Every sequence by family, and how the data was verified |
docs/img/ | The screenshots of this README |
favicon.svg, favicon.ico, apple-touch-icon.png | The site icon (light theme), at the root where browsers look for it |
CNAME, .nojekyll | GitHub Pages: custom domain; serve files as they are |
data/catalog.csv (RFC 4180, header row): id (folder and URL name), anumber, name, alias,
family, keywords (the OEIS keywords core and nice), n0 (first index = OEIS offset), terms, chunks, chunk_rows, the counts
decomposable, level, weight, ties, level_one, forced, the ranges amin … dmax,
and the note.
Each chunk is gzip-compressed binary; every number is an unsigned LEB128 varint:
| Part | Contents |
|---|
dwj2 | format tag (4 bytes) |
m | 1 byte: jumps stored as 0 d, 1 first differences, 2 second differences (zigzag) |
n, a0 | rows in the chunk, first term |
j[0…n−1] | the jumps, coded by m |
s[0…n−1] | the smaller factor of a − d = k·L: 0 no decomposition, 2k if k ≤ L, 2L + 1 if L < k |
Every term is present, so row i is index n0 + i. The page rebuilds a as a running sum of the
jumps and the other factor as (a − d) divided by the stored one, and rejects a chunk if anything
is inconsistent. All values are below 2⁵³. tools/compact.py has encode() and decode() for
Python; older dwj1 chunks are still read. A sequence weighs from under 1 kB to 0.6 MB (median
0.16 MB); the gzip streams are made by Zopfli, about 4 % smaller than gzip -9.
Rebuilding the data
The pipeline is deterministic: unchanged sequences produce no diff. It needs a C compiler, Python
3.9+ with numpy, Pillow and zopfli, ~2 GB of RAM, ~13 GB of disk and about two hours of CPU.
cd tools
cc -O2 -o decompwlj_gen decompwlj_gen.c -lm
./decompwlj_gen raw # every sequence: raw a,d,k,L chunks + catalog.csv (~60 min)
./decompwlj_gen raw primes 200000 # or one sequence, at any length below 2^53
python3 names.py raw # OEIS names into the catalogue (from oeis.json)
python3 audit.py raw # independent check of every row and of the OEIS terms
python3 compact.py raw ../data # the compact chunks the site loads
python3 thumbs.py raw ../thumbs # the gallery plates
python3 seo.py # static pages, share images, sitemap, robots.txt
To add a sequence: write a generator in tools/decompwlj_gen.c (it fills t[0..cnt-1] with
strictly increasing terms), add its entry to the defs[] table, record its OEIS data with
python3 fetch_oeis.py A123456, then run the pipeline; audit.py must report "all clear".
Data integrity
The numbers are protected at every step, from the generator to the file a visitor saves:
- At build time,
tools/verify_all.c, a program that shares no code with the generator
(its own decoder and its own factorisation), checks every row of every sequence:
a = k·L + d exactly, k > d, k is the least divisor of a − d above d, a term decomposes
exactly when a > 2d, the chunks are well formed and continue each other, the catalogue's
counts are reproduced, and the first terms equal the OEIS terms. The last full run checked
499,965,535 rows and re-derived 499,947,311 weights in under 8 minutes, with no failure.
- Fingerprints:
data/sha256.txt holds the SHA-256 of every sequence's CSV, computed by
an independent decoder (tools/fingerprint.py), in sha256sum format; each sequence page
shows its own.
- In the browser, every chunk is checked as it is decoded (exact divisions, k > d, the
stored factor, row counts, continuity), then the viewer recounts the catalogue's figures and
compares the SHA-256 of the sequence with the published one: the Sequence panel says
"✓ data verified" or shows a warning. Both CSV buttons compare the file's SHA-256 with the
published one and refuse to save a file that differs.
- The graphs: every gallery plate is redrawn from the data and compared pixel by pixel
(
tools/check_derived.py); in the browser test, sixty sequences are loaded in the viewer and
every point drawn (6 million) must sit exactly at log k, log L, log d of its term, in its class.
- The OEIS links: every A-number is well formed and unique, its OEIS record (
tools/oeis.json)
has the catalogue's name and an offset at or before the data's first index, its first terms equal
the data's, and the viewer and every sequence page link to https://oeis.org/<A-number>.
- Afterwards, anyone can check downloaded CSVs:
sha256sum -c sha256.txt.
- Continuously: every pull request checks all of the above for all 5000 sequences: every
row, every fingerprint, every plate, every OEIS link.
.github/workflows/integrity.yml repeats
it every week.
The pipeline above ends, when data change, with:
cc -O2 -o tools/verify_all tools/verify_all.c -lz && python3 tools/verify_all.py # every row
python3 tools/fingerprint.py # data/sha256.txt
python3 tools/seo.py # pages carry it
Automatic checks
Every pull request and push to main runs checks.yml, in
parallel jobs so that checking everything takes no longer than sampling did:
- rows 0/8 … 7/8: eight runners, an eighth of the sequences each:
verify_all on every row,
and check_derived.py on every CSV fingerprint and every plate;
- site: the generator compiles with
-Werror; tools/check_site.py finds every file in place,
checks every OEIS link and rebuilds four sequences byte for byte; audit.py re-verifies one
sequence in 25 from the site's own chunks; seo.py must rewrite the same HTML;
- data: passes when the rows and the site pass;
- browser:
tests/smoke.mjs drives Chromium through the gallery, the viewer, sixty sequences
(every point drawn checked against its term), a CSV download and the How it works page.
Locally:
python3 tools/check_site.py && python3 tools/audit.py data --every 25
python3 tools/check_derived.py # every fingerprint and plate (about 7 min on 4 cores)
python3 -m http.server 8765 & node tests/smoke.mjs # needs: npm install playwright
Troubleshooting
| Symptom | Fix |
|---|
| Blank page / "Open this page through a web server" | Serve it over HTTP (see Run it locally). |
| Old version after an update | Hard reload (Ctrl+F5, Cmd+Shift+R); Pages can take a few minutes. |
| "Enforce HTTPS" greyed out | DNS not pointing at GitHub yet, a leftover A record or redirection, or a CAA record blocking Let's Encrypt. Fix, then remove and re-add the custom domain. |
| A sequence fails to load (404 on a chunk) | The server rewrites missing files, or data/ was not fully uploaded. |
| Gallery works, 3-D view does not | WebGL is off (some VMs and remote desktops): enable hardware acceleration. |
Credits and licences
- The decomposition into weight × level + jump: Rémi Eismann —
arXiv:0711.0865, decompwlj.com.
- The decomposition data (
data/: a, d, k, L for every term, computed by the programs in tools/):
Rémi Eismann, dedicated to the public domain under
CC0 1.0.
- The decompositions were computed, and this site built, with Claude (Anthropic's AI model) through
Claude Code, under the direction of Rémi Eismann.
- Sequence names, offsets and reference terms: the OEIS, CC BY-SA 4.0, read
from github.com/oeis/oeisdata.
- three.js r169, © 2010–2024 three.js authors, MIT (
vendor/LICENSE-three.js.txt).
- mp4-muxer 5.2.2, © 2023 Vanilagy, MIT (
vendor/LICENSE-mp4-muxer.txt).
