THE FOLD / RESPAWN / SECOND WIND / THE EKG SEQUENCE
THE EKG SEQUENCE
a sequence walking by shared factors
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The EKG sequence (its plot looks like a heartbeat trace) grows by a single greedy rule: start 1, 2, and each next term is the smallest positive integer not yet used that shares a common factor with the previous term. It runs 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, … It is conjectured (and largely proven) to be a permutation of all positive integers — every number appears exactly once. A proven structural fact: every prime p first appears immediately after 2p, and is immediately followed by 3p.
LIT verified live: over the first 3000 terms, consecutive terms always share a factor > 1, all terms are distinct, every integer 1..1000 appears, and each interior prime p is preceded by 2p and followed by 3p (window.__ekg). FIG honest: full permutation is proven in the literature; here the defining rule and the 2p/3p structure are checked over a finite prefix.
LIT verified live: over the first 3000 terms, consecutive terms always share a factor > 1, all terms are distinct, every integer 1..1000 appears, and each interior prime p is preceded by 2p and followed by 3p (window.__ekg). FIG honest: full permutation is proven in the literature; here the defining rule and the 2p/3p structure are checked over a finite prefix.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at second-wind — each step revives the smallest number left unspent that still shares a factor with where you are; the sequence keeps finding a way onward. That revival is the second wind. AVAN (AI) built the instrument: the greedy smallest-unused-with-shared-factor rule, the gcd-chain check, the distinctness and coverage checks, and the 2p-before / 3p-after prime structure.
Credit as content: the EKG sequence (Jonathan Ayres; analysed by Lagarias, Rains & Sloane, 2002; OEIS A064413). The weave: David names second-wind; I extend the sequence by always taking the least unused number sharing a factor with the last, then confirm consecutive terms are never coprime, nothing repeats, the small integers all arrive, and every prime sits between 2p and 3p.
Credit as content: the EKG sequence (Jonathan Ayres; analysed by Lagarias, Rains & Sloane, 2002; OEIS A064413). The weave: David names second-wind; I extend the sequence by always taking the least unused number sharing a factor with the last, then confirm consecutive terms are never coprime, nothing repeats, the small integers all arrive, and every prime sits between 2p and 3p.
3 ONE DIMENSION
1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, … each term shares a factor with the last. 7 (prime) arrives right after 14 = 2·7 and is followed by 21 = 3·7.
4 TWO DIMENSIONS · INTERACTIVE
The sequence as a heartbeat trace; the gcd-chain, distinctness, coverage, and 2p/3p prime rule checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a walk that never runs out of shared factors.
AVAN’s addition (the inverse-companion): order the integers not by size but by who shares a factor with whom — always step to the smallest unused neighbour in the divisibility graph. The inverse of ‘count 1, 2, 3, 4’ is ‘walk the integers by common factors, taking the least available each time.’ Magenta is the plain counting order; green is the shared-factor EKG walk. The same integers, reordered by their factors.
LIT Genuine EKG sequence (Jonathan Ayres; analysed by Jeffrey Lagarias, Eric Rains & N.J.A. Sloane, 2002; OEIS A064413). Verified live: the greedy 'smallest unused sharing a factor' rule generates 3000 terms where consecutive terms are never coprime (window.__ekg.gcdChain), all terms are distinct (window.__ekg.distinct), 1..1000 all appear (window.__ekg.covers), and every interior prime p is preceded by 2p and followed by 3p (window.__ekg.structural).
FIG No framing: the greedy generation, the gcd-chain check, distinctness, coverage, and the 2p/3p prime structure all run in-browser with exact arithmetic. Honest scope: the full permutation-of-the-integers property is proven in the literature; here the defining rule and the proven 2p/3p structure are checked over a finite 3000-term prefix (primes appear late, so coverage is bounded at 1..1000 below the frontier). The AVAN inverse is honest — ordering the integers by shared factors rather than by size (least available neighbour in the divisibility graph) genuinely reorders them; magenta is the plain counting order, green the shared-factor EKG walk. The same integers, reordered by their factors.
FIG No framing: the greedy generation, the gcd-chain check, distinctness, coverage, and the 2p/3p prime structure all run in-browser with exact arithmetic. Honest scope: the full permutation-of-the-integers property is proven in the literature; here the defining rule and the proven 2p/3p structure are checked over a finite 3000-term prefix (primes appear late, so coverage is bounded at 1..1000 below the frontier). The AVAN inverse is honest — ordering the integers by shared factors rather than by size (least available neighbour in the divisibility graph) genuinely reorders them; magenta is the plain counting order, green the shared-factor EKG walk. The same integers, reordered by their factors.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN