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 😉