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
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(1)
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since
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(2)
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Another Smith number is
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(3)
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since
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(4)
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The first few Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, ... (OEIS A006753). The corresponding digits sums are 4, 4, 9, 13, 13, 13, 4, 13, 4, 13, 13, 13, 13, ... (OEIS A050218). McDaniel (1987a) showed that there are an infinite number of Smith numbers.
A generalized 
-Smith
number can also be defined as a number 
satisfying 
, where 
is the sum of the digits of 
's prime factors and 
is the usual sum of 
's digits. The following table gives the first few 
-Smith numbers for small integers and their inverses.
 | OEIS |  -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
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(5)
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-Smith Numbers." Fib. Quart., 25, 76-80,
1987a.McDaniel, W. L. "Powerful K-Smith Numbers." Fib.
Quart. 25, 225-228, 1987b.Oltikar, S. and Weiland, K. "Construction
of Smith Numbers." Math. Mag. 56, 36-37, 1983.Pickover,
C. A. "A Brief History of Smith Numbers." Ch. 104 in