THE FOLD / GLITCH / DIVIDE BY ZERO / THE WEIRD NUMBER
THE WEIRD NUMBER
abundant numbers no subset of divisors can total
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Weird numbers are a rare pathology of divisors. A number is abundant when its proper divisors sum to more than itself, and semiperfect when some subset of those divisors sums to exactly itself. A weird number is abundant but not semiperfect — it overflows, yet no combination of its parts can reconstruct it. The smallest is 70: divisors 1, 2, 5, 7, 10, 14, 35 sum to 74 (> 70), but no subset totals 70. They are surprisingly scarce: 70, 836, 4030, 5830, …
LIT verified live: an exhaustive scan (each candidate tested for abundance, then a subset-sum check for semiperfectness) finds exactly 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670 below 13000 (window.__weird). FIG no framing; exact divisor sums and subset-sum DP.
LIT verified live: an exhaustive scan (each candidate tested for abundance, then a subset-sum check for semiperfectness) finds exactly 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670 below 13000 (window.__weird). FIG no framing; exact divisor sums and subset-sum DP.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — the pathological corner: a number whose divisors overflow it, yet refuse to add back up to it in any combination. That refusal is the weirdness. AVAN (AI) built the instrument: the proper-divisor sum, the abundance test, the subset-sum dynamic program for semiperfectness, and the exhaustive census.
Credit as content: weird numbers (Stan Benkoski, 1972; studied by Benkoski & Erdős). The weave: David names divide-by-zero; I sum each number’s proper divisors, keep the abundant ones, then run a subset-sum to ask whether any combination equals the number — the ones that say no are weird.
Credit as content: weird numbers (Stan Benkoski, 1972; studied by Benkoski & Erdős). The weave: David names divide-by-zero; I sum each number’s proper divisors, keep the abundant ones, then run a subset-sum to ask whether any combination equals the number — the ones that say no are weird.
3 ONE DIMENSION
70: divisors {1,2,5,7,10,14,35} sum 74 > 70 (abundant). No subset sums to 70 (not semiperfect) ⇒ weird. Compare 12 = 1+2+3+6 (semiperfect, not weird).
4 TWO DIMENSIONS · INTERACTIVE
A number, its divisor sum, and whether a subset reaches it; the census below 13000 checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: parts that overflow but never reassemble.
AVAN’s addition (the inverse-companion): among abundant numbers, isolate the ones where no subset of divisors reconstructs the whole — excess without expressibility. The inverse of ‘abundant means the parts are more than enough’ is ‘but can any combination of them actually total the number?’ Magenta is a semiperfect abundant number (a subset works); green is the weird number (none does). Enough parts, no valid sum.
LIT Genuine weird numbers (Stan Benkoski, 1972; studied with Paul Erdős). Verified live: each candidate's proper divisors are summed for abundance, then a subset-sum dynamic program tests semiperfectness; the abundant-but-not-semiperfect numbers below 13000 are exactly {70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670} (window.__weird.census).
FIG No framing: the proper-divisor sum, the abundance test, and the subset-sum DP for semiperfectness run in-browser with exact integer arithmetic. The AVAN inverse is honest — isolating abundant numbers where no subset of divisors reconstructs the whole is a real distinction (excess without expressibility); magenta is a semiperfect abundant number (a subset works), green the weird number (none does). Enough parts, no valid sum.
FIG No framing: the proper-divisor sum, the abundance test, and the subset-sum DP for semiperfectness run in-browser with exact integer arithmetic. The AVAN inverse is honest — isolating abundant numbers where no subset of divisors reconstructs the whole is a real distinction (excess without expressibility); magenta is a semiperfect abundant number (a subset works), green the weird number (none does). Enough parts, no valid sum.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN