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
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). There are360 Smith numbers less than
and 29,928 
. McDaniel (1987a) showed that an infinite number exist.A generalized 
-Smith number can also be defined as a number 
satisfying 
, where 
is the sum of primefactors and 
is the sum of digits. There are 47 1-Smith numbers, 21 2-Smith numbers, three 3-Smith numbers, and one7-Smith, 9-Smith, and 14-Smith number 
.
A Smith number can be constructed from every factored Repunit 
. The largest known Smith number is
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 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. SequenceA006753/M3582in ``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.-Smith Numbers.'' Fib. Quart., 25, 76-80, 1987a.
- Johnson's Equation
- Johnson Solid
- Johnson's Theorem
- Join (Graph)
- Join (Spaces)
- Joint Distribution Function
- Joint Probability Density Function
- Joint Theorem
- Jonah Formula
- Jones Polynomial
- Jonquière's Function
- Jordan Algebra
- Jordan Curve
- Jordan Curve Theorem
- Jordan Decomposition Theorem
- Jordan-Hölder Theorem
- Jordan Measure
- Jordan Polygon
- Jordan's Inequality
- Jordan's Lemma
- Josephus Problem
- Jug
- Jugendtraum
- Juggling
- Julia Fractal