200 Follower Problem

There is an easy criterion for determining whether a positive integer is faithful.

Lemma. A positive integer $n$ is faithful if and only if it has a divisor $a$ such that $n/a - 1$ is a composite number greater than $4$ .

Proof. By definition, a positive integer $n$ is faithful if and only if there exist integers $a$ , $r$ , and $s$ with $r,s \geq 2$ such that $a + ar + ars = n \iff a[1 + r(1 + s)] = n \iff r(1 + s) = \frac na - 1.$ Since $r,s \geq 2$ , the left side of equation on the right is simply a general representation for a composite number greater than $4$ .

We claim that all integers $n > 40$ are faithful. To show this, choose the largest value $k \in \{0,1,2,3,4\}$ such that $n = 2^k d$ for some integer $d$ . If $k \leq 3$ , then $d$ is odd since otherwise we could have chosen a larger value of $k$ . Put $a = 2^k$ , and then $\frac 1a \geq \frac 18 \implies d = \frac na > \frac {40}8 = 5 \implies d - 1 = \frac na - 1 > 4.$ Since $d$ is odd in this case, the relation on the right shows $n/a - 1$ is even and greater than $4$ and thus composite. If instead $k = 4$ , then put $a = d$ , and then $\frac na - 1 = 2^4 - 1 = 15,$ which is a composite number greater than $4$ . In either case, $n/a - 1$ is composite and greater than $4$ , so $n$ is faithful based on our lemma.

Now we have shown that we only need to consider positive integers less than or equal to $40$ to find all the non-faithful numbers, and the lemma is a convenient enough way to check for faithfulness. Nine numbers turn out not to be faithful, $\text{1, 2, 3, 4, 5, 6, 8, 12, 24},$ and the sum of these numbers is $\boxed{65}$ .