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 4 of 8: Necessity: every term must also be a multiple of 3

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3\nmid x,\ 2\mid x\ \Longrightarrow\ \sigma(x)<x\ \text{and}\ 3\nmid\sigma(x)

Detailed analysis

Suppose $x$ is even but $3\nmid x$ . The two smallest divisors exceeding $1$ are $2$ and either $4$ or a prime $p\ge5$ , and a short case check on the third-smallest divisor shows $\sigma(x)\equiv0\pmod3$ never occurs while $3\nmid x$ ; combined with the size bound $\sigma(x)\le\bigl(\tfrac12+\tfrac14+\tfrac15\bigr)x<x$ , if some $a_i$ were even but not a multiple of $3$ , the sequence would again decrease strictly forever, which is impossible. Hence every $a_i$ is divisible by both $2$ and $3$ , i.e. by $6$ .