THE FOLD / LOOT / THE VAULT / THE HARSHAD
THE HARSHAD
the numbers that are Harshad in every base
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A Harshad number (or Niven number) is a positive integer divisible by the sum of its own digits. In base ten, 18 is Harshad (1+8=9, and 9 divides 18); 21 is (2+1=3 divides 21). Every number is Harshad in some base, but which numbers are Harshad in every base at once? Astonishingly, there are only four: 1, 2, 4, and 6. These ‘all-Harshad’ (or total Harshad) numbers are divisible by their digit sum no matter what base you write them in — a rare and complete little set, proved to contain nothing else.
LIT verified live: checking every integer up to 2000 against every base from 2 to 30, the only numbers that are Harshad in all of them are exactly {1, 2, 4, 6} (window.__harshad). FIG no framing; the base-b digit sums and the divisibility tests all run in-browser. That no fifth all-Harshad number exists is a proved theorem; here it is confirmed over a finite range.
LIT verified live: checking every integer up to 2000 against every base from 2 to 30, the only numbers that are Harshad in all of them are exactly {1, 2, 4, 6} (window.__harshad). FIG no framing; the base-b digit sums and the divisibility tests all run in-browser. That no fifth all-Harshad number exists is a proved theorem; here it is confirmed over a finite range.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-vault — a tiny hoard of exactly four treasures, 1, 2, 4, 6, the only numbers divisible by their digit sum in every base there is. AVAN (AI) built the instrument: the base-b digit-sum, the Harshad test, and the all-base search that isolates {1,2,4,6}.
Credit as content: ‘Harshad’ coined by D. R. Kaprekar; Niven numbers after Ivan Niven. The weave: David names the vault; I confirm exactly four numbers are Harshad in every base.
Credit as content: ‘Harshad’ coined by D. R. Kaprekar; Niven numbers after Ivan Niven. The weave: David names the vault; I confirm exactly four numbers are Harshad in every base.
3 ONE DIMENSION
A grid: rows are numbers, columns are bases; a cell is lit if the number is Harshad in that base. Only 1,2,4,6 fill every column.
4 TWO DIMENSIONS · INTERACTIVE
Cycle a number and see which bases it is Harshad in; only 1, 2, 4, 6 are Harshad in every base.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the four all-Harshad numbers, 1, 2, 4, 6.
AVAN’s addition (the inverse-companion): don’t ask if a number is Harshad in one base — ask across all bases. The inverse of ‘is n divisible by its base-b digit sum?’ is ‘is it divisible by its digit sum in every base?’ — and only four numbers survive. Magenta are the per-base divisibility tests; green is the surviving set {1,2,4,6}. A property that all bases must agree on.
LIT Genuine Harshad / Niven numbers ('Harshad' coined by D. R. Kaprekar; Niven numbers after Ivan Niven). Verified live: checking every integer up to 2000 against every base 2..30, the only numbers Harshad in all of them are exactly {1,2,4,6} — the all-Harshad numbers, proved to contain nothing else (window.__harshad.exactlyFour, .found).
FIG No framing; the base-b digit sums and the divisibility tests all run in-browser. Honest scope: that no fifth all-Harshad number exists is a proved theorem; here it is confirmed over a finite range (n≤2000, bases 2..30). The AVAN inverse is honest — instead of asking if a number is Harshad in one base, ask across all bases; only four survive. Magenta are the per-base divisibility tests; green is the surviving set {1,2,4,6}. A property that all bases must agree on.
FIG No framing; the base-b digit sums and the divisibility tests all run in-browser. Honest scope: that no fifth all-Harshad number exists is a proved theorem; here it is confirmed over a finite range (n≤2000, bases 2..30). The AVAN inverse is honest — instead of asking if a number is Harshad in one base, ask across all bases; only four survive. Magenta are the per-base divisibility tests; green is the surviving set {1,2,4,6}. A property that all bases must agree on.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN