Enter a composite integer to check if it is a Smith number (digit sum equality)
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Result
Verdict
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Sum of Digits (n)
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Sum of Digits (factors)
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Detailed Derivation
Smith Number Definition
Composite n where digit_sum(n) = digit_sum(prime_factors(n))
Prime factors counted with multiplicity
Smallest Smith number: 4 = 2x2 (4 = 2+2)
Most famous: 4937775 (Smith's phone number)
A Smith number is a composite integer whose digit sum equals the sum of the digits of its prime factors. These numbers were discovered accidentally when mathematician Albert Wilansky noticed his brother-in-law Smith's phone number had this property.
⚠Prime numbers are excluded by definition. The sum includes all prime factor digits with repetition (multiplicity counted).
What Is a Smith Number?
A Smith number is a composite where the sum of digits equals the sum of digits of its prime factors. Discovered by Albert Wilansky in 1982, these numbers are named after his brother-in-law whose phone number (493-7775) was the first such number identified.
Digit Sum
Add all digits of the number. 666: 6+6+6=18. 4937775: 4+9+3+7+7+7+5=42.
Factor Digit Sum
Factor the number and add digits of each prime factor (with repetition). 666=2x3x3x37: 2+3+3+3+7=18.
Discovery
Wilansky noticed 4937775 had equal digit sums. He named them after his brother-in-law Smith. The number factors as 3x5x5x65837.
Infinite
Proved infinitely many by McDaniel (1987). Over 29,000 exist below 1 million. They appear irregularly but consistently.
Teaching Example: Test 666. Factor: 666 = 2 x 3 x 3 x 37. Sum digits of 666: 6+6+6=18. Sum factor digits: 2+3+3+3+7=18. Equal! 666 is a Smith number. Also try 4, 22, 27, 121, 666, 4937775.
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