Blast From The Past

Looking back over my old tweets on Twitter (as it was once called), I noticed that my first reference to a day count occurred on August 11th 2013. See Figure 1.

Figure 1

It's a fairly unimpressive tweet that, from a mathematical perspective, simply notes that 23506 has four distinct prime factors. I wasn't aware of the Online Encyclopedia of Integer Sequences or OEIS back then, not Numbers Aplenty or most other resources that I now use. The OEIS informs us that 23506 is a member of A029793:

 A029793    Numbers $k$ such that $k$ and $k^2$ have the same set of digits.

This is because $23506^2=552532036$ and there are not many numbers in the range up to 40000 that have this property. The numbers that do are: 

0, 1, 10, 100, 1000, 4762, 4832, 10000, 10376, 10493, 11205, 12385, 12650, 14829, 22450, 23506, 24605, 26394, 34196, 36215

From Numbers Aplenty we learn that 23506 has properties that make it

• a self number because there is no number that added to its sum of digits gives 23506
  
• an untouchable number because it is not equal to the sum of proper divisors of any number

My own algorithm tells me that 23506 is:

• a xenodrome in base 9: 35217 and in base 10: 23506 because all digits are different
  
• an attractor because its sum of even and odd digits are the same (8)

These are just a few of the special properties that 23506 possesses. I've come a long way since those early days.