Can You Measure Up?

Riddler Classic

From Emma Knight comes a unit conversion conundrum:

The sum of the factors of 36 — including 36 itself — is 91. Coincidentally, 36 inches rounded to the nearest centimeter is … 91 centimeters!

Can you find another whole number like 36, where you can “compute” the sum of its factors by converting from inches to centimeters?

Extra credit: Can you find a third whole number with this property? How many more whole numbers can you find?

Solution

I originally applied guess and check to some numbers which I thought would be in the realm of $\sigma(n) > 2n$. Two of those numbers, $24$ and $132$ were off by 1.

I ran some code on the first 10,000,000 numbers.

There are only three numbers that satisfy the problem, three numbers that are off by 1, and four numbers that are off by 2.

Answer

$378$ is the next number after $36$.

Extra credit:

$49600$ is the only other number I could determine. This is also the only number in the table greater than 510. I am speculating these numbers are extremely rare, and perhaps nonexistent for > 10,000,000.

Rohan Lewis

2021.11.07

Code can be found here.