Smith Number

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

$$666=2\cdot 3\cdot 3\cdot 37,$$

since

$$6+6+6=2+3+3+(3+7)=18.$$

Another Smith number is

$$4937775=3\cdot 5\cdot 5\cdot 65837,$$

since

$$4+9+3+7+7+7+5=3+5+5+(6+5+8+3+7)=42.$$

The first few Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, ... (Sloane's A006753). There are 360 Smith numbers less than $10^4$ and 29,928 $\leq 10^6$. McDaniel (1987a) showed that an infinite number exist.

A generalized $k$-Smith number can also be defined as a number $m$ satisfying $S_p(m)=kS(m)$, where $S_p$ is the sum of prime factors and $S$ is the sum of digits. There are 47 1-Smith numbers, 21 2-Smith numbers, three 3-Smith numbers, and one 7-Smith, 9-Smith, and 14-Smith number $<1000$.

A Smith number can be constructed from every factored Repunit $R_n$. The largest known Smith number is

$$9\times R_{1031}(10^{4594}+3\times 10^{2297}+1)^{1476}\times 10^{3913210}.$$

See also Monica Set, Perfect Number, Repunit, Smith Brothers, Suzanne Set

References

Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 99-300, 1989.

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

McDaniel, W. L. ``The Existence of Infinitely Many $k$-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.

Sloane, N. J. A. Sequence A006753/M3582 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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.