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 5 of 8: Three formulas for multiples of 6

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6\mid x:\quad \sigma(x)=\tfrac{13}{12}x\ (4\mid x),\qquad \sigma(x)=\tfrac{31}{30}x\ (4\nmid x,\,5\mid x),\qquad \sigma(x)=x\ (4\nmid x,\,5\nmid x)

Detailed analysis

Every term now satisfies $6\mid x$ , so $2,3,6$ are always divisors; the three smallest divisors exceeding $1$ depend on whether $4$ or $5$ also divide $x$ . If $4\mid x$ , they are $2,3,4$ , giving $\sigma(x)=\tfrac{x}2+\tfrac{x}3+\tfrac{x}4=\tfrac{13}{12}x$ . If $4\nmid x$ but $5\mid x$ , they are $2,3,5$ , giving $\sigma(x)=\tfrac{x}2+\tfrac{x}3+\tfrac{x}5=\tfrac{31}{30}x$ . If $4\nmid x$ and $5\nmid x$ , they are $2,3,6$ , giving $\sigma(x)=\tfrac{x}2+\tfrac{x}3+\tfrac{x}6=x$ .