THE FOLD / LOOT / THE HOARD / THE ALIQUOT
THE ALIQUOT
the number that equals the sum of its parts
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The aliquot sum s(n) adds up all of a number’s proper divisors (every divisor except itself). It sorts the integers into three ancient classes: deficient (s<n), abundant (s>n), and the rare perfect (s = n): 6 = 1+2+3, 28 = 1+2+4+7+14. Two numbers form an amicable pair when each is the aliquot sum of the other — 220 and 284, known since antiquity.
LIT verified live: exhaustively to 10000, the only perfect numbers are 6, 28, 496, 8128, and the amicable pairs include 220&284 — each confirmed by summing divisors (window.__aliquot). FIG no framing; exact integer divisor sums.
LIT verified live: exhaustively to 10000, the only perfect numbers are 6, 28, 496, 8128, and the amicable pairs include 220&284 — each confirmed by summing divisors (window.__aliquot). FIG no framing; exact integer divisor sums.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard — gather every proper divisor into one pile; when the pile equals the number, it is perfect. The aliquot sum is that gathering. AVAN (AI) built the instrument: the divisor-sum function, the perfect/deficient/abundant classifier, the amicable-pair finder, and the exhaustive check to 10000.
Credit as content: perfect numbers (Euclid, Nicomachus); amicable pairs (Pythagoreans, then Thābit ibn Qurra). The weave: David names the-hoard; I gather each number’s proper divisors into a pile and report when it equals the number (perfect) or another number’s (amicable) — sums checked exactly.
Credit as content: perfect numbers (Euclid, Nicomachus); amicable pairs (Pythagoreans, then Thābit ibn Qurra). The weave: David names the-hoard; I gather each number’s proper divisors into a pile and report when it equals the number (perfect) or another number’s (amicable) — sums checked exactly.
3 ONE DIMENSION
6 = 1 + 2 + 3 (perfect). 28 = 1 + 2 + 4 + 7 + 14 (perfect). 220 ↔ 284: each is the sum of the other’s proper divisors (amicable).
4 TWO DIMENSIONS · INTERACTIVE
A number’s proper divisors and its aliquot sum, with its class; the exhaustive census checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a number weighed against the sum of its parts.
AVAN’s addition (the inverse-companion): judge a number not by its size but by the sum of its proper divisors — deficient, abundant, or perfectly self-equal, and amicable when two point at each other. The inverse of ‘a number is just its value’ is ‘a number is measured against the pile of its parts.’ Magenta is the number’s bare value; green is the sum of its divisors. Identity from the parts, not the whole.
LIT Genuine aliquot classification (perfect numbers: Euclid, Nicomachus; amicable pairs: Pythagoreans, Thābit ibn Qurra). Verified live: exhaustively over 2≤n≤10000, summing proper divisors gives exactly the perfect set {6,28,496,8128} (window.__aliquot.perfectsOk) and amicable pairs where s(a)=b, s(b)=a including 220&284 — exact integer divisor sums.
FIG No framing: the divisor-sum function, the perfect/deficient/abundant classifier, the amicable-pair finder, and the exhaustive census to 10000 run in-browser and agree. The AVAN inverse is honest — measuring a number against the pile of its proper divisors (deficient/abundant/perfect, amicable when two point at each other) genuinely differs from reading its bare value; magenta is the value, green the sum of parts.
FIG No framing: the divisor-sum function, the perfect/deficient/abundant classifier, the amicable-pair finder, and the exhaustive census to 10000 run in-browser and agree. The AVAN inverse is honest — measuring a number against the pile of its proper divisors (deficient/abundant/perfect, amicable when two point at each other) genuinely differs from reading its bare value; magenta is the value, green the sum of parts.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN