When you plot the "importance" N(n) of the positive integers, where
N(n) = The number of occurrences of n in the OEIS
an interesting anomaly appears[1]. The "importance" of the integers fall into two clusters, separated by Sloane's Gap[2][3].
Also, Neil Sloane appeared[4] several times on Numberphile. In the last few videos of that[4] playlist ("Amazing Graphs"), Sloane shares some of the particularly interesting sequences from the OEIS.
You can try to get a rough idea by doing web searches for various random integers of different lengths (courtesy of random.org, which graced me with lots of 7's).
7 - A lucky number
92 - How old Lady Gaga wants to be when she retires
420 - A class of dinghy
7260 - A wireless card
5108 - A section in the U.S. Code about supplemental claims
51664 - A hydraulic filter
794101 - A postal code in India
7578146 - A patent number of a knitting loom clip
74703567 - A clothing trademark serial number for "authentic bad lands"
Were Raymond Smullyan still with us, he would point out that you just answered your own question. And yet didn't. In which case you did. Provided you didn't.
Another example:
"The smallest integer that has a description too long to fit in a Hacker News comment."
But you can definitely say "what is the meta information or algorithm to compute X". E.g. Pi is an irrational number, but we have a bunch of ways to specify pi (we just can't explicitly write down the full sequence of numbers). Another interesting thing, while the full sequence is not computable, any particular precision IS computable (ie for any number Y that you can write down exactly, you can compute the Y'th digit of pi).
These sorts of self-referential statements are actually forbidden in the axioms of Zermelo-Fraenkel set theory because they require unrestricted comprehension [1]. ZF set theory specifically restricted comprehension to avoid Russell’s paradox [2] as well as countless other statements, like these, which lead to absurdities in math.
But it is more than fun to explore them, as Smullyan points out. For example, it leads to discussions like, “What is a description?” Which is near and dear to my heart, as it leads to “What is a program?” and, “What is the specification of the machine that runs the program?”
I’ve been having a fantastic time in my philosophy of math course this term. It’s incredible how deep and how long these debates have been running. Cantor, Frege, Russell, Hilbert, Heyting, Gödel, Quine, and on and on!
I forget which book is the source of this, but I recall Smullyan writing about (I hope I have it roughly right) asking a child whether they could prove something they knew about mathematics or logic, and the child replied "What is a proof?"
Smullyan said that this was--if you took it literally--an incredibly deep question.
10k characters might be a browser limit, but I tried pasting 9,000 characters from one of my essays into a comment, and it was also rejected as being too long.
More interestingly, the set of integers seems to be split into two sets: those that occur relatively often, and those that occur relatively rarely (same page)
It is hard to overestimate the importance of the OEIS in enumerative combinatorics.
I discovered the main results of my PhD thesis essentially as follows:
1. Find complicated construction A, hoping to prove some new results.
2. Fail to sufficiently understand/analyze A.
3. Write computer program to analyze characteristics of A for small n.
4. Using OEIS, discover that apparently A is (in some sense) equivalent to some completely different construction B, which is much simpler and well-understood.
5. Show desired result as well as further other results using B and variations of it.
Just 3 weeks ago someone uploaded a video of Neil Sloane talking about his encyclopedia. At his 80th birthday he is as vivid and enthusiastic as always.
I still remember being excited when I got a sequence accepted, and even more excited to get a second sequence accepted. In each case it was something I worked on, but had been given by someone else.
Also, Neil Sloane appeared[4] several times on Numberphile. In the last few videos of that[4] playlist ("Amazing Graphs"), Sloane shares some of the particularly interesting sequences from the OEIS.
[1] https://i.stack.imgur.com/vcNYb.png
[2] https://arxiv.org/abs/1101.4470v2
[3] https://www.youtube.com/watch?v=_YysNM2JoFo
[4] https://www.youtube.com/playlist?list=PLt5AfwLFPxWJXQqPe_llz...