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.

[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...

Many previous threads: https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...

(On HN, reposts are ok after a year or so: https://news.ycombinator.com/newsfaq.html)

An episode of Numberphile from back in August¹ had a great interview with the creator. His enthusiasm is /still/ huge.

1. https://www.numberphile.com/podcast/neil-sloane

I haven't checked the podcast, but the YouTube channel has uploaded some videos featuring him pretty recently, which I found to be very amusing.

A fun question I have been pondering recently: what is the smallest positive integer that has not yet been written or uttered by a human being?

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"

760314276 - No useful search results.

2641309883 - No useful search results

24235991397 - One (useless) search result

415865230233 - Zero search results

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.

Fun stuff anyway!

[1] https://en.wikipedia.org/wiki/Axiom_schema_of_specification#...

[2] https://en.wikipedia.org/wiki/Russell%27s_paradox

Absolutely!

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?”

Or some of my favourites:

“What is a number? Do numbers exist?”

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.

What is the limit? This <textarea> element does not specify one, nor do I see any limit mentioned in the text of this web page.

I want to post an extremely long comment and see what the limit is now…

It seems to be 10k chars.

Otherwise, it displays a "message too long" error.

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.

Reminds me of this:

https://en.m.wikipedia.org/wiki/Interesting_number_paradox

If you limit it to mathematicians, it was, at some time, 8795 ;-) (https://oeis.org/wiki/Frequency_of_appearance_in_the_OEIS_da...)

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)

Every time you answer the question the correct answer changes.

Side effects in math!!!!!!!!!!

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.

There is even an old competition on Kaggle to try to predict the next integer of the given sequence.

https://www.kaggle.com/c/integer-sequence-learning

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.

https://vimeo.com/365825314

It has Maple and Mathematica code, but nothing for open source alternatives, e.g. Octave, Julia or Python. See exampe[1].

[1] https://oeis.org/search?q=1%2C2%2C3%2C6%2C11%2C23%2C47%2C106...

Not “nothing”. That page has PARI code. PARI is GPL licensed (https://pari.math.u-bordeaux.fr/)

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.

Even so, happy days ...

Twitter bot that tweets out random visually appealing integer sequences: https://twitter.com/amazing_graphs

The EIS is so valuable it should be designated a Treasure of the World Heritage of Mankind by UNESCO.