# 2023

Random fact about 2023: if you divide 2023 by the sum of its digits (in base 10), you get a square number:

$$\displaystyle \frac{2023}{2+0+2+3} = \frac{2023}{7} = 289 = 17^2$$.

This is sequence A001102 in the On-Line Encyclopedia of Integer Sequences (OEIS). The last year that satisfied this property was 1815; however, OEIS doesn’t say when it next applies. Let’s find out.

First we need to spell out the property so that it’s easier to automate a search.

Let $$d_x(i)$$ denote the $$i$$th digit of $$x$$ in base 10,

$$\displaystyle d_x(i) = \frac{n \bmod 10^{i+1}-n \bmod 10^i}{10^i}$$,

where $$i$$ ranges from $$0$$ to $$\lfloor \log_{10}{x} \rfloor$$, i.e., to one less than the number of digits of $$x$$.

Let $$\displaystyle j(k) = \frac{k}{\sum_{i = 0}^{ \lfloor \log_{10}{k} \rfloor} d_k(i)}$$.

The property holds of $$k \in \mathbb{N}$$ iff there is an $$m \in \mathbb{N}$$ such that $$j(k) = m^2$$, or, equivalently, if $$\lfloor \sqrt{j(k)} \rfloor^2=j(k)$$.

This is enough to code up the search in R. Here are the next few years when the property applies: 2023, 2025, 2028, 2178, 2304, 2312, 2352, 2400.

Please note, I’m not a numerologist, so I don’t know if this is a good thing; however, Happy New Year nonetheless 😉