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 3 of 8: Necessity: every term must be even

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x\ \text{odd}\ \Longrightarrow\ \sigma(x)<x\ \text{and}\ \sigma(x)\ \text{odd}

Detailed analysis

If $x$ is odd, every divisor of $x$ is odd, so $\sigma(x)$ , a sum of three odd numbers, is odd; moreover the three smallest divisors exceeding $1$ are at least $3,5,7$ , so $\sigma(x)\le\bigl(\tfrac13+\tfrac15+\tfrac17\bigr)x<x$ . If some $a_i$ were odd, the sequence would be odd and strictly decreasing from that point on forever, which is impossible for an infinite sequence of positive integers. Hence every $a_i$ is even.