Smith Numbers

A Smith number is a composite number, the sum of whose digits equals the sum of the digits of all its prime factors.

The smallest Smith number is 4 = 2×2. The sum of the digits of 4 is 4, and The sum of digits of its prime factors is 2 + 2 = 4.

Another example is 6,036 = 2×2×3×503; 6,036 has a digit sum of 15, and the sum of the digits of its prime factors is also 15.

How many Smith numbers are there between 2 and 10,000? ... In 1987 Wayne McDaniel showed that infinitely many Smith numbers exist.

Patrick Costello & Kathy Lewis in Mathematics Magazine (June 2002); "Lots of Smiths," Vol. 75, No. 3. (Jun., 2002), pp. 223-226.

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There are 376 Smith numbers less than 10,000.

The first ten are 4, 22, 27, 58, 85, 94, 121, 166, 202, and 265.

The last three are 9,942; 9,975; and 9,985.

"Albert Wilansky noticed Dr. Harold Smith's phone number 493-7775 when written as the single number 4937775 ... is a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. Adding the digits in the number and the digits of its prime factors 3, 5, 5, and 65837 resulted in identical sums of 42."

[ 4937775 = 3(5)(5)(65837)

4+9+3+7+7+7+5 = 42 = 3+5+5+6+5+8+3+7; see "Lots of Smiths"] Another such number found by him is 6036.

W. McDaniel, The existence of infinitely many k-Smith numbers, Fibonacci Quarterly 25 (1987), 76-80.
Density issues with Smith's could be explored [email Don Dechman for a QBasic computer program: dondechman@verizon.net]. A comparison with other numbers' - e.g., Primes, Ulam numbers) - asymptotic density may be of some interest.
 7/10/2008 Version http://www.cs.ucla.edu/~klinger/nmath/smith.html ©2008 Allen Klinger