An infinite sequence $a_1,a_2,\dots$ of positive integers is defined as follows: each $a_n$ has at least three proper divisors, and $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$. Determine all possible values of $a_1$.

Step 1: A proper divisor of $n$ is any positive divisor $<n$. The three largest proper divisors of $n$ are $n/p_1,n/p_2,n/p_3$ where $p_1<p_2<p_3$ are the three smallest prime factors of $n$ (if $n$ has enough factors).

Step 2 (Example: $a_1=12$): Divisors of 12: 1, 2, 3, 4, 6. Proper: 1, 2, 3, 4, 6. Three largest: 6, 4, 3. Sum = 13. But 13 is prime — only proper divisor is 1. Doesn't have 3 proper divisors. So $a_1=12$ fails.

Step 3 (Example: $a_1=6$): Proper divisors: 1, 2, 3. Three largest: 3, 2, 1. Sum = 6. So $a_2=6$: the sequence is constant. Works!

Step 4: Characterize all $a_1$ for which the sequence never reaches a number with fewer than 3 proper divisors. This requires $a_n$ to always have at least 3 proper divisors.

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