Construction of Smith Number from Prime Repunit

Theorem
Let $R_n$ be a repunit which is prime where $n \ge 3$.

Then $3304 \times R_n$ is a Smith number.

$3304$ is not the only number this works for, but it is the smallest.

Proof
Let $\map S n$ denote the sum of the digits of a positive integer $n$.

Let $\map {S_p} n$ denote the sum of the digits of the prime decomposition of $n$.

Then $\map S n = \map {S_p} n$ $n$ is a Smith number.

Let $n \ge 3$.

We have that:
 * $3304 = 2 \times 2 \times 2 \times 7 \times 59$

and so for a prime repunit $R_n$:
 * $3304 \times R_n = 2 \times 2 \times 2 \times 7 \times 59 \times \underbrace {111 \ldots 11}_{n \text { ones} }$

and so:
 * $\map {S_p} {3304 \times R_n} = 2 + 2 + 2 + 7 + 5 + 9 + n \times 1 = n + 27$

Note that:

This gives:

Hence the result.