Smith number


In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the given number base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed.
Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number of his brother-in-law Harold Smith:
while
in base 10.

Mathematical definition

Let be a natural number. For base, let the function be the digit sum of n in base. A natural number has the integer factorisation
and is a Smith number if
where is the p-adic valuation of.
For example, in base 10, 378 = 21 33 71 is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 21 111 is a Smith number, because 2 + 2 = 2 · 1 + · 1
The first few Smith numbers in base 10 are:

Properties

W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers.
The number of Smith numbers in base 10 below 10n for n=1,2,... is:
Two consecutive Smith numbers are called Smith brothers. It is not known how many Smith brothers there are. The starting elements of the smallest Smith n-tuple in base 10 for n = 1, 2,... are:
Smith numbers can be constructed from factored repunits. The largest known Smith number in base 10 is:
where R1031 is a repunit equal to /9.