A Smith number is a composite number the sum of whose digits is the sum of the digits of its prime factors (excluding 1). (The primes are excluded since they trivially satisfy this condition). One example of a Smith number is the beast number
since
Another Smith number is
since
The first few Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, ... (Sloane's A006753). The corresponding digits sums are 4, 4, 9, 13, 13, 13, 4, 13, 4, 13, 13, 13, 13, ... (Sloane's A050218). McDaniel (1987a) showed that there are an infinite number of Smith numbers.
A generalized k-Smith number can also be defined as a number m satisfying 
,
is the sum of the digits of m's prime factors and S(m) is the usual sum of m's digits. The following table gives the first few k-Smith numbers for small integers and their inverses.
| k | Sloane | k-Smith numbers |
 |
A050225 | 6969, 19998, 36399, 39693, 66099, 69663, ... |
 |
A050224 | 88, 169, 286, 484, 598, 682, 808, 844, 897, ... |
| 1 | A006753 | 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, ... |
| 2 | A104390 | 32, 42, 60, 70, 104, 152, 231, 315, 316, 322, ... |
| 3 | A104391 | 402, 510, 700, 1113, 1131, 1311, 2006, 2022, ... |
A Smith number can be constructed from every factored repunit 
(Hoffman 1998, pp. 205-206). The largest known Smith number is
Hoax Number, Monica Set, Perfect Number, Repunit, Smith Brothers, Suzanne Set
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Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 205-206, 1998.
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McDaniel, W. L. "Powerful K-Smith Numbers." Fib. Quart. 25, 225-228, 1987b.
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Sloane, N. J. A. Sequences A006753/M3582, A050218, A050224, A050225, A104390, and A104391 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
Wilansky, A. "Smith Numbers." Two-Year College Math. J. 13, 21, 1982.
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