# Smith number

In 1982, when Albert Wilansky called his brother-in-law, he noticed that the phone number was composite and that the sum of the digits in the phone number equals the sum of the digits in its prime factors.

4937775 = 3.5.5.65837
4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7

Composite numbers with this property are now called Smith numbers after the brother-in-law Wilansky was calling. Trivially, all prime numbers have this property, so they are excluded. The Smith numbers less than 1000 are:

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, and 985.

In 1987, Wayne McDaniel showed that are infinitely many Smith numbers by constructing a sequence of them.

If Rn is a repunit prime, then 1540.Rn is a Smith number (with digital sum 18+n). Note that 1540 is not the only possible mutiplier here, others include:

1540, 1720, 2170, 2440, 5590, 6040, 7930, 8344, 8470, 8920, 23590, 24490, 25228, 29080, 31528, 31780, 33544, 34390, 35380, 39970, 40870, 42490, 42598, 43480, 44380, 45955, 46270, 46810, 46990, 47908, 48790, and 49960.

Related pages (outside of this work)

References:

Lewis1986
K. Lewis, "Smith numbers: an infinite subset of N," Master's thesis, M.S., Eastern Kentucky University, (1994)
McDaniel87
W. McDaniel, "The existence of infinitely many k-Smith numbers," Fibonacci Quart., 25 (1987) 76--80.  MR 88d:11007
McDaniel87b
W. McDaniel, "Palindromic Smith numbers," J. Recreational Math., 19:1 (1987) 34--37.
OW83
S. Oltikar and K. Wayland, "Construction of Smith numbers," Math. Mag., 56 (1983) 36--37.  MR 84e:10015
Wilansky82
A. Wilansky, "Smith numbers," Two-Year College Math. J., 13 (1982) 21.
Yates1991
S. Yates, "Welcome back, Dr. Matrix," J. Recreational Math., 23:1 (1991) 11--12.
Yates82
S. Yates, Repunits and repetends, Star Publishing Co., Inc., 1982.  Boynton Beach, Florida, pp. vi+215, MR 83k:10014