A composite integer N whose digit sum S(N) is equal to the sum of the
digits of its prime factors Sp (N) is called a Smith number [17].
For example 85 is a Smith number because digit sum of 85
(i.e. S(85) = 8 + 5=13), which is equal to the sum of the digits of its
prime factors i.e. Sp (85) = Sp (17 x 5) = 1 + 7 + 5 = 13.
Albert Wilansky named Smith numbers from his brother-in-law Herald
Smith's telephone number 4937775 with this property
(i.e. 4937775 = 3.5.5.65837)
Since
4+9+3+7+7+7+5=3+5+5+(6+5+8+3+7)=42
Wilansky also mentioned two other numbers with this property
(i.e. 9985 and 6036). Wilansky has found that there are 360 Smith
numbers less than 10000, which is not correct, as there are 376 Smith
numbers less then 10000. It is now known that there are infinitely many
Smith numbers [2].
There are 25154060 smith numbers below 109[4]. Further computations
reveal that there are 241882509 smith numbers below 1010.
All 376 Smith numbers below 10000 are:
004, 022, 027, 058, 085, 094, 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, 985,
1086, 1111, 1165, 1219, 1255, 1282, 1284, 1376, 1449, 1507, 1581, 1626,
1633, 1642, 1678, 1736, 1755, 1776, 1795, 1822, 1842, 1858, 1872, 1881,
1894, 1903, 1908, 1921, 1935, 1952, 1962, 1966, 2038, 2067, 2079, 2155,
2173, 2182, 2218, 2227, 2265, 2286, 2326, 2362, 2366, 2373, 2409, 2434,
2461, 2475, 2484, 2515, 2556, 2576, 2578, 2583, 2605, 2614, 2679, 2688,
2722, 2745, 2751, 2785, 2839, 2888, 2902, 2911, 2934, 2944, 2958, 2964,
2965, 2970, 2974, 3046, 3091, 3138, 3168, 3174, 3226, 3246, 3258, 3294,
3345, 3366, 3390, 3442, 3505, 3564, 3595, 3615, 3622, 3649, 3663, 3690,
3694, 3802, 3852, 3864, 3865, 3930, 3946, 3973, 4054, 4126, 4162, 4173,
4185, 4189, 4191, 4198, 4209, 4279, 4306, 4369, 4414, 4428, 4464, 4472,
4557, 4592, 4594, 4702, 4743, 4765, 4788, 4794, 4832, 4855, 4880, 4918,
4954, 4959, 4960, 4974, 4981, 5062, 5071, 5088, 5098, 5172, 5242, 5248,
5253, 5269, 5298, 5305, 5386, 5388, 5397, 5422, 5458, 5485, 5526, 5539,
5602, 5638, 5642, 5674, 5772, 5818, 5854, 5874, 5915, 5926, 5935, 5936,
5946, 5998, 6036, 6054, 6084, 6096, 6115, 6171, 6178, 6187, 6188, 6252,
6259, 6295, 6315, 6344, 6385, 6439, 6457, 6502, 6531, 6567, 6583, 6585,
6603, 6684, 6693, 6702, 6718, 6760, 6816, 6835, 6855, 6880, 6934, 6981,
7026, 7051, 7062, 7068, 7078, 7089, 7119, 7136, 7186, 7195, 7227, 7249,
7287, 7339, 7402, 7438, 7447, 7465, 7503, 7627, 7674, 7683, 7695, 7712,
7726, 7762, 7764, 7782, 7784, 7809, 7824, 7834, 7915, 7952, 7978, 8005,
8014, 8023, 8073, 8077, 8095, 8149, 8154, 8158, 8185, 8196, 8253, 8257,
8277, 8307, 8347, 8372, 8412, 8421, 8466, 8518, 8545, 8568, 8628, 8653,
8680, 8736, 8754, 8766, 8790, 8792, 8851, 8864, 8874, 8883, 8901, 8914,
9015, 9031, 9036, 9094, 9166, 9184, 9193, 9229, 9274, 9276, 9285, 9294,
9296, 9301, 9330, 9346, 9355, 9382, 9386, 9387, 9396, 9414, 9427, 9483,
9522, 9535, 9571, 9598, 9633, 9634, 9639, 9648, 9657, 9684, 9708, 9717,
9735, 9742, 9760, 9778, 9840, 9843, 9849, 9861, 9880, 9895, 9924, 9942,
9968, 9975, 9985.
Note that the Beast number 666 is also a Smith Number.