THE FOLD / LOOT / THE VAULT / THE AMICABLE
THE AMICABLE
two numbers summing to each other's divisors
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Amicable numbers are two different numbers, each of which equals the sum of the other’s proper divisors. The smallest pair is (220, 284): the proper divisors of 220 (1,2,4,5,10,11,20,22,44,55,110) sum to 284, and the proper divisors of 284 (1,2,4,71,142) sum to 220. They point at each other perfectly. Known since Pythagoras and prized by mystics as a symbol of friendship, they generalize the perfect numbers (where a number is amicable with itself). The next pair is (1184, 1210), then (2620, 2924).
LIT verified live: with s(n) = σ(n) - n (the sum of proper divisors), s(220) = 284 and s(284) = 220; a brute search confirms (220, 284) is the smallest amicable pair, and the next is (1184, 1210) (window.__amicable). FIG no framing; the divisor sums and the pair search both run in-browser.
LIT verified live: with s(n) = σ(n) - n (the sum of proper divisors), s(220) = 284 and s(284) = 220; a brute search confirms (220, 284) is the smallest amicable pair, and the next is (1184, 1210) (window.__amicable). FIG no framing; the divisor sums and the pair search both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-vault — the loot: a rare matched pair of numbers, each holding the other’s divisor-sum, locked together like two keys to one vault. AVAN (AI) built the instrument: the proper-divisor sum, the mutual test, and the smallest-pair search.
Credit as content: known to the Pythagoreans (220, 284); Thabit ibn Qurra’s rule for generating pairs. The weave: David names the vault; I confirm 220 and 284 sum to each other’s divisors.
Credit as content: known to the Pythagoreans (220, 284); Thabit ibn Qurra’s rule for generating pairs. The weave: David names the vault; I confirm 220 and 284 sum to each other’s divisors.
3 ONE DIMENSION
220 and 284, each pointing at the other: the divisors of 220 sum to 284, and vice versa.
4 TWO DIMENSIONS · INTERACTIVE
Cycle amicable pairs; s(a)=b and s(b)=a are checked from the proper-divisor sums.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the amicable pair, each the other's divisor-sum.
AVAN’s addition (the inverse-companion): don’t judge a number alone — sum its divisors and follow the arrow. The inverse of ‘the number a’ is ‘s(a), the sum of its proper divisors’; when s(a)=b and s(b)=a, the two are amicable. Magenta are the proper divisors; green is the partner each pair sums to. Two numbers holding each other’s divisors.
LIT Genuine amicable numbers (known to the Pythagoreans for (220,284); Thabit ibn Qurra's generating rule). Verified live: s(220)=284 and s(284)=220 (s=sum of proper divisors); a brute search confirms (220,284) is the smallest amicable pair and (1184,1210) the next (window.__amicable.s220, .s284, .first, .second, .ok).
FIG No framing; the divisor sums and the pair search both run in-browser. The AVAN inverse is honest — instead of judging a number alone, sum its divisors and follow the arrow: the inverse of 'the number a' is 's(a), the sum of its proper divisors'; when s(a)=b and s(b)=a, the two are amicable. Magenta are the proper divisors; green is the partner each pair sums to. Two numbers holding each other's divisors.
FIG No framing; the divisor sums and the pair search both run in-browser. The AVAN inverse is honest — instead of judging a number alone, sum its divisors and follow the arrow: the inverse of 'the number a' is 's(a), the sum of its proper divisors'; when s(a)=b and s(b)=a, the two are amicable. Magenta are the proper divisors; green is the partner each pair sums to. Two numbers holding each other's divisors.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN