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 8 of 8: Conclusion

Previous step Next step

a_1=12^{e}\cdot6\ell,\qquad e,\ell\ge0\ \text{integers},\ \gcd(\ell,10)=1

Detailed analysis

Step 6 shows every valid $a_1$ has the form $12^e\cdot6\ell$ with $\gcd(\ell,10)=1$ (taking $e=T-1\ge0$ ), and step 2 shows every such value indeed generates a valid infinite sequence. Hence the complete answer is $a_1=12^e\cdot6\ell$ for integers $e,\ell\ge0$ with $\gcd(\ell,10)=1$ .