Publishing mathematical coincidences

There is actually the possibility for publishable research on this topic, in the context of computational complexity: How many formulas should one try for a relative accuracy of $10^{-p}$? The answer is $O(10^{p/2})$, see Seemingly Remarkable Mathematical Coincidences Are Easy to Generate by William Press (2009).$^\ast$

How hard is it to find such coincidences? Must one try on the order of $10^{19}$ formulas before getting one as good as equation (4)? _{[whose error is only about $2 \times 10^{−19}$]} Not at all. The standard cryptographic technique of "meet-in-the-middle attack" yields a computational complexity of order the square root of the inverse precision, requiring also about this much memory. So coincidences as good as equation (4) can readily be found in $O(10^9)$ operations on an ordinary desktop machine with $O(10^9)$ memory. The rest of this note gives the details of one such implementation.

_{$^\ast$ As far as I could check, this article was not published in a journal, but the point of my answer is that it could have been.}

You answered your question yourself: it is published on the web site that you refer to. The author of the web site cites his sources, in most cases these are personal communications. Many results of this sort are spread by correspondence, on Internet, and by oral personal communication. There is also a journal "Experimental Mathematics" which publishes these kinds of observations, even those for which the author has no explanation.

A few of those that pop up on the internet every now and then (for instance on xkcd) are actually found using RIES, a program made for finding algebraic approximations given a decimal (possibly rounded) representation, by Robert Munafo.

This makes it easy to find such coincidences and many of those have been published. But these usually fall more under recreational or experimental math, as they are not appearing in the context of some "serious" research.