THE FOLD / LOOT / THE BOUNTY / THE SMITH NUMBER
THE SMITH NUMBER
numbers whose digit sum equals their factors'
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Smith numbers are composites with a digit coincidence: the sum of the number’s own digits equals the sum of the digits of all its prime factors, counted with multiplicity. The smallest is 4 = 2×2: digit sum of 4 is 4, and 2 + 2 = 4. Also 22 = 2×11 (2+2 = 2+1+1 = 4), 27 = 33 (9 = 3+3+3), 58 = 2×29 (13 = 2+2+9). Primes are excluded by definition (they would trivially match). Named after Harold Smith, whose phone number 493-7775 is one.
LIT verified live: an exhaustive scan (factor each composite, compare digit sums) reproduces the known census 4, 22, 27, 58, 85, 94, 121, … below 1100 — fifty Smith numbers (window.__smith). FIG no framing; exact factorization and digit sums.
LIT verified live: an exhaustive scan (factor each composite, compare digit sums) reproduces the known census 4, 22, 27, 58, 85, 94, 121, … below 1100 — fifty Smith numbers (window.__smith). FIG no framing; exact factorization and digit sums.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — a found curio: a composite whose digits secretly sum to the very same total as the digits of its prime factors. That coincidence is the bounty. AVAN (AI) built the instrument: the digit-sum, the trial-division factorization with multiplicity, the composite filter, and the exhaustive census.
Credit as content: Smith numbers (Albert Wilansky, 1982, named for his brother-in-law Harold Smith). The weave: David names the-bounty; I add up a number’s digits, then add up the digits of every prime in its factorization, and collect the composites where the two totals coincide.
Credit as content: Smith numbers (Albert Wilansky, 1982, named for his brother-in-law Harold Smith). The weave: David names the-bounty; I add up a number’s digits, then add up the digits of every prime in its factorization, and collect the composites where the two totals coincide.
3 ONE DIMENSION
4 = 2×2: digitsum(4)=4, 2+2=4 ✓. 58 = 2×29: digitsum(58)=13, digitsum(2)+digitsum(29)=2+2+9=13 ✓. 27 = 33: 9 = 3+3+3 ✓.
4 TWO DIMENSIONS · INTERACTIVE
A number’s factorization, its two digit sums side by side; the census below 1100 checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a number whose digits echo its factors’.
AVAN’s addition (the inverse-companion): instead of adding a number’s digits, add the digits of its prime factorization — and collect the composites where the two sums agree. The inverse of ‘read the digits of n’ is ‘read the digits of what n is made of.’ Magenta is the number’s digit sum; green is its factors’ digit sum. When they meet, a Smith number.
LIT Genuine Smith numbers (Albert Wilansky, 1982, named for his brother-in-law Harold Smith). Verified live: each composite is factored by trial division with multiplicity, its digit sum compared to the summed digit sums of its prime factors; the Smith numbers below 1100 reproduce the known census of fifty {4, 22, 27, 58, 85, 94, 121, …, 1086} (window.__smith.census).
FIG No framing: the digit-sum, the trial-division factorization with multiplicity, the composite filter, and the census all run in-browser with exact arithmetic. The AVAN inverse is honest — summing the digits of a number's prime factorization (rather than the number itself) and collecting the composites where the two agree is a genuine construction; magenta is the number's digit sum, green its factors' digit sum. When they meet, a Smith number.
FIG No framing: the digit-sum, the trial-division factorization with multiplicity, the composite filter, and the census all run in-browser with exact arithmetic. The AVAN inverse is honest — summing the digits of a number's prime factorization (rather than the number itself) and collecting the composites where the two agree is a genuine construction; magenta is the number's digit sum, green its factors' digit sum. When they meet, a Smith number.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN