Problem 4

A proper divisor of a positive integer

N

is a positive divisor of

N

other than

N

itself. The infinite sequence

a_1,a_2,\ldots

consists of positive integers, each of which has at least three proper divisors. For each

n\ge1

, the integer

a_{n+1}

is the sum of the three largest proper divisors of

a_n

. Determine all possible values of

a_1

Step 1 of 8: Setup: the three largest proper divisors

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\sigma(x):=\text{sum of the three largest proper divisors of }x,\qquad a_{n+1}=\sigma(a_n)

Detailed analysis

Let $S$ be the set of positive integers with at least three proper divisors, i.e. at least four divisors in total. For $x\in S$ , the three largest proper divisors of $x$ are $x/d_1,x/d_2,x/d_3$ where $d_1<d_2<d_3$ are the three smallest divisors of $x$ exceeding $1$ ; write $\sigma(x)$ for their sum, so the recursion is $a_{n+1}=\sigma(a_n)$ .