Smith Number

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

(1)

since

(2)

Another Smith number is

(3)

since

(4)

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

(5)

REFERENCES:

Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 99-100, 1989.

Guy, R. K. "Smith Numbers." §B49 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 103-104, 1994.

Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, pp. 205-206, 1998.

McDaniel, W. L. "The Existence of Infinitely Many -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 Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England: Oxford University Press, pp. 247-248, 2001.

Sloane, N. J. A. Sequences A006753/M3582, A050218, A050224, A050225, A104390, and A104391 in "The On-Line Encyclopedia of Integer Sequences."

Wilansky, A. "Smith Numbers." Two-Year College Math. J. 13, 21, 1982.

Yates, S. "Special Sets of Smith Numbers." Math. Mag. 59, 293-296, 1986.

Yates, S. "Smith Numbers Congruent to 4 (mod 9)." J. Recr. Math. 19, 139-141, 1987.