THE FOLD / RESPAWN / GARBAGE COLLECTION / THE ULAM NUMBERS
THE ULAM NUMBERS
a sequence that builds itself from unique sums
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ulam numbers are a sequence that builds itself. Start with 1 and 2. Each new term is the smallest integer larger than the last that can be written as a sum of two distinct earlier Ulam numbers in exactly one way. That single rule generates 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, … — 3 = 1+2, 4 = 1+3, but 5 is excluded (5 = 1+4 = 2+3, two ways). Devised by Stanislaw Ulam in 1964, the sequence looks random yet has a startling hidden regularity: its terms cluster around a nearly-constant density, and a mysterious ‘almost period’ of about 21.6 governs where they fall — still not fully explained.
LIT verified live: the self-generating rule reproduces the known Ulam sequence exactly — the first 26 terms match 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, …, 99 — and each term has exactly one representation as a sum of two distinct earlier terms (window.__ulam). FIG no framing; the sequence is generated from the rule and checked in-browser.
LIT verified live: the self-generating rule reproduces the known Ulam sequence exactly — the first 26 terms match 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, …, 99 — and each term has exactly one representation as a sum of two distinct earlier terms (window.__ulam). FIG no framing; the sequence is generated from the rule and checked in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at garbage-collection — the respawn: the sequence keeps regenerating itself, each new term summoned from the unique-sum rule over all that came before. AVAN (AI) built the instrument: the generator, the exactly-one-way test, and the match to the known sequence.
Credit as content: Stanislaw Ulam (1964). The weave: David names the self-regeneration; I confirm the rule reproduces the Ulam numbers exactly.
Credit as content: Stanislaw Ulam (1964). The weave: David names the self-regeneration; I confirm the rule reproduces the Ulam numbers exactly.
3 ONE DIMENSION
The Ulam numbers on a line; each new one is the smallest with a unique two-term sum from earlier terms.
4 TWO DIMENSIONS · INTERACTIVE
Step through terms; each is shown with its unique representation, and 5 is shown excluded (two ways).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: each Ulam number, admitted by its unique sum.
AVAN’s addition (the inverse-companion): don’t list numbers and test them — let the set decide who joins. The inverse of ‘the next integer’ is ‘the smallest with exactly one representation as a sum of two earlier members’. Magenta are the two earlier terms that sum to it; green is the Ulam number they uniquely admit. A sequence that selects its own members.
LIT Genuine Ulam numbers (Stanisław Ulam, 1964). Verified live: the self-generating rule reproduces the known sequence exactly — the first 26 terms match 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, …, 99 — and each term has exactly one representation as a sum of two distinct earlier terms (window.__ulam.ok).
FIG No framing; the sequence is generated from the rule and checked in-browser. The AVAN inverse is honest — instead of listing numbers and testing them, let the set decide who joins: the inverse of 'the next integer' is 'the smallest with exactly one representation as a sum of two earlier members'. Magenta are the two earlier terms that sum to it; green is the Ulam number they uniquely admit. A sequence that selects its own members.
FIG No framing; the sequence is generated from the rule and checked in-browser. The AVAN inverse is honest — instead of listing numbers and testing them, let the set decide who joins: the inverse of 'the next integer' is 'the smallest with exactly one representation as a sum of two earlier members'. Magenta are the two earlier terms that sum to it; green is the Ulam number they uniquely admit. A sequence that selects its own members.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN