Special Sets of Smith Numbers
Samuel Yates · Mathematics Magazine · 1986
Wilansky [1] defined a Smith number to be a composite number such as 4 = 2 x 2, 22 = 2 x 11, or 27 = 3 X 3 X 3, the sum of whose digits is the same as the sum of the digits in its representation as a product of primes. Professor Wilansky's brother-in-law, H. Smith, had a telephone number, 4937775 = 3 X 5 x 5 x 65837, which is a Smith number, whence the name. Oltikar and Wayland [2] found larger Smith numbers, including one with 362 digits. In this note, I find much larger ones, including one with 7,158 digits. Generally, integers of this size are notoriously difficult to factor, but you will see that if you have large primes of certain forms, it is easy to find large Smith numbers from them. We can get an idea of the sparseness of Smith numbers from Wilansky's count of them. He found that there are 47 among the first 1,000 nonnegative integers, and between 28 and 42 in each of the next nine 1,000-integer intervals, the average there being 35.