The Connoisseur of Number Sequences

For more than 50 years, the mathematician Neil Sloane has curated the authoritative collection of interesting and important integer sequences.


Neil Sloane curates the Online Encyclopedia of Integer Sequences (OEIS) from his attic home office in Highland Park, N.J. (Photo: John Smock for Quanta Magazine)


Formerly a children’s playroom, the attic is filled with giant stacks of papers containing number sequence submissions from all over the world. (Photo: John Smock for Quanta Magazine)


The OEIS, which began in 1964 as a stack of handwritten index cards, now contains more than a quarter of a million different sequences of numbers that arise in different mathematical contexts. (Photo: John Smock for Quanta Magazine)


Over the course of a more than 40-year research career at Bell Labs (later AT&T Labs), the 75-year-old Welsh native has won numerous awards for papers in the fields of combinatorics, coding theory, optics and statistics. (Photo: John Smock for Quanta Magazine)


Sloane has personally created entries for more than 170,000 sequences. (Photo: John Smock for Quanta Magazine)

Neil Sloane is considered by some to be one of the most influential mathematicians of our time.

That’s not because of any particular theorem the 75-year-old Welsh native has proved, though over the course of a more than 40-year research career at Bell Labs (later AT&T Labs) he won numerous awards for papers in the fields of combinatorics, coding theory, optics and statistics. Rather, it’s because of the creation for which he’s most famous: the Online Encyclopedia of Integer Sequences (OEIS), often simply called “Sloane” by its users.

This giant repository, which celebrated its 50th anniversary last year, contains more than a quarter of a million different sequences of numbers that arise in different mathematical contexts, such as the prime numbers (2, 3, 5, 7, 11 … ) or the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13 … ). What’s the greatest number of cake slices that can be made with n cuts? Look up sequence A000125 in the OEIS. How many chess positions can be created in n moves? That’s sequence A048987. The number of ways to arrange n circles in a plane, with only two crossing at any given point, is A250001. That sequence just joined the collection a few months ago. So far, only its first four terms are known; if you can figure out the fifth, Sloane will want to hear from you.

A mathematician whose research generates a sequence of numbers can turn to the OEIS to discover other contexts in which the sequence arises and any papers that discuss it. The repository has spawned countless mathematical discoveries and has been cited more than 4,000 times.

“Many mathematical articles explicitly mention how they were inspired by OEIS, but for each one that does, there are at least ten who do not mention it, not necessarily out of malice, but because they take it for granted,” wrote Doron Zeilberger, a mathematician at Rutgers University.

Courtesy of Neil Sloane

The number of ways to arrange n circles in a plane, with only two crossing at any given point, is sequence A250001 in the OEIS.

The collection, which began in 1964 as a stack of handwritten index cards, gave rise to a 1973 book containing 2,372 sequences, and then a 1995 book, co-authored with mathematician Simon Plouffe, containing just over 5,000 sequences. By the following year, so many people had submitted sequences to Sloane that the collection nearly doubled in size, so he moved it onto the Internet. Since then, Sloane has personally created entries for more than 170,000 sequences. Recently, however, he’s had help processing the torrent of submissions he receives each year from all over the world: Since 2009 the collection has been run as a wiki, and it now boasts more than 100 volunteer editors.

But the OEIS is still very much Sloane’s baby. He spends hours each day vetting new submissions and adding sequences from archived papers and correspondence.

Quanta caught up with Sloane over Skype last month as he sorted through sequences in his attic home office in Highland Park, N.J. Formerly a children’s playroom, its garish wallpaper is tempered by giant stacks of papers, and, as Sloane put it, “enough computers so I don’t need a heater.” An edited and condensed version of the interview follows.

QUANTA MAGAZINE: Tell me how you started the OEIS. Some sequences came up in your research as a graduate student, right?

NEIL SLOANE: It was my thesis. I was looking into what are now called neural networks. These are networks of [artificial] neurons, and each neuron fires or doesn’t fire and is connected to other neurons which fire or don’t fire depending on the signal. I wanted to know whether the activity in some of these networks was likely to die out or keep firing.

Some of the simplest cases gave rise to sequences. I took the simplest one and, with some difficulty, worked out half a dozen terms. [It] goes 1, 8, 78, 944…. I needed to know how fast it grew, and I looked it up in the obvious places, and it wasn’t there.

I started making a collection of sequences, so the next time this came up, I’d have my own table to look up. I made a little collection of file cards, and then they became punched cards and then magnetic tape and eventually the book in 1973.

And when did you start sharing your collection with other people?

Oh, right away. I mean, within a year or two. The word got around, and you know, letters started coming in. And as soon as the book came out, there was a flood of letters. I’m still going through the binders from that period. The project [now] is to sort through all the interesting documents from the past, which now goes back 51 years. A lot of them are in binders. A lot of them are not, unfortunately. Over there, there’s about an eight- or nine-feet stack of papers that haven’t been sorted.

It’s very slow work. I have to go through these 50 binders and figure out what’s worth scanning, what’s worth preserving, what is available online so we don’t need to scan it. But I’m also finding lots of new sequences as I go along, that for one reason or another I didn’t include the first time around.

Besides the books about sequences, you’ve also co-authored two guidebooks to rock climbing in New Jersey.

I did it with my climbing partner, Paul Nick. We spent a lot of time driving around New Jersey climbing on crags and taking photographs and collecting route information. There were a lot of restrictions. A lot of cliffs were on private property, so we couldn’t officially include them in the book.

Do you have any favorite mathematical discoveries that came about because of the OEIS?

One of the most famous discoveries has to do with a formula discovered by Gregory, an astronomer back in Newton’s day, for π/4. The formula says that π/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 and so on. It’s a good way of computing π if you don’t have any better way. So somebody did this, but wondered what would happen if you stopped after a while. So he truncated the sum after 500,000 terms and looked at the number, and he worked it out to many decimal places. He noticed, of course, that it was different from π.

John Smock for Quanta Magazine

Sloane is discovering new integer sequences in unsorted stacks of documents collected over 51 years.

He looked at where it differed, and it differed after five decimal places. But then it agreed for the next ten places, and then it disagreed for two decimal places. Then it agreed for the next ten places, and then it disagreed. This was absolutely amazing, that it would agree everywhere except at certain places.

Then I think it was Jonathan Borwein who looked at the differences [between π and the truncated sum]. When you subtract you get a sequence of numbers, and he looked it up in the OEIS, and it wasn’t there. But then he divided by 2 and looked it up, and there they were. It was sequence A000364. It was the Euler numbers.

He and his two collaborators studied this, and they ended up with a formula for the error term. If you truncate Gregory’s series after not just 500,000 terms, but after n terms, where n can be anything you want, you can give an exact formula for the error.

It was absolutely miraculous that this was discovered. So, it’s a theorem that came into existence because of the OEIS.

Tell me about some sequences you like. What makes a sequence appealing to you?

It’s a bit like saying, “What makes a painting appealing?” or “What makes a piece of music appealing?” In the end, it’s just a matter of judgment, based on experience. If there is some rule for generating the sequence which is a bit surprising, and the sequence turns out to be not so easy to understand, that makes it interesting.

There’s a sequence of Leroy Quet’s which produces primes. It chugs along, but it’s like Schrödinger’s cat; we don’t know if it exists [as an infinitely long sequence] or not. I think we’ve computed 600 million terms, and so far it hasn’t died. It would be nicer — or maybe it would be less nice — if we could actually analyze it.

How often do you get a new sequence that makes you say, “I can’t believe no one has ever thought of this before”?

This happens all the time. There are many gaps, even now. I fill in these gaps myself quite often when I come across something in one of these old letters. We’re a finite community. It’s easy to overlook even an obvious sequence.

To what extent is there a clear aesthetic about which sequences deserve to be in the OEIS?

We have arguments about this, of course, because somebody will send in a sequence that he or she thinks is wonderful, and we the editors, look at it and say, “Well, that’s really not very interesting. That’s boring.” Then the person who submitted it may get really annoyed and say, “No, no, you’re wrong. I spent a lot of time on this sequence.” It’s a matter of judgment, and in the end I have the final say. Of course, I’m very influenced by the other editors-in-chief.

One of our phrases is, “This is too specialized. This is too arbitrary. This is not of general interest.” For instance, primes beginning with 1998 would not be so interesting. Too specialized, too arbitrary, so that would be rejected.

It might not be rejected if it had been published somewhere — if it was on a test, say. We like to include sequences that appear on IQ tests. It’s always been one of my goals to help people do these silly tests.

One of the features on the OEIS is the option to listen to a sequence musically. What do you think that adds?

Well, it’s another dimension of looking at the sequence. Some sequences, you get a good feeling for them by listening to them. Some of the sequences almost sound like music. Others just sound like rubbish.

You’ve said that you think Bach would have loved the OEIS.

I think music is very mathematical, obviously, and so he would have appreciated the OEIS. He would have understood it. He probably would have joined in, contributed some sequences. Maybe he would have composed some pieces that we could use.

Do you have a sense of the magnitude of the OEIS’ impact?

Not really. I know it’s helped a lot of people, and it’s very famous. We have sequence fans from all over the world. You’ll see many references from unexpected places to the OEIS: journals, books, theses from civil engineering or social studies that mention sequences. They come up all over the place.

Are there other repositories of mathematical information that you wish existed, but don’t yet?

You would like an index to theorems, but it’s hard to imagine how that would work.

We’re trying to get a collaboration going with the Zentralblatt — the German equivalent of Math Reviews’ MathSciNet — about making it possible to search for formulas in the OEIS. Suppose you want the summation of xn over n2 + 3, where the sum goes from one to infinity. It’s very hard to look that up in the OEIS at present.

You’re retired from AT&T Labs, but looking at your list of recent publications and your activity with the OEIS, you seem anything but retired.

I have an office at Rutgers, and I give lectures there, and I have students, and I’m even busier back here in my study running the OEIS and doing research and going around the world giving talks and so on. I’m busier than ever.

There are more than 4,000 people registered on the OEIS website. They range from professional mathematicians to recreational mathematicians, right?

A child just registered the other day, and said, “I’m ten years old, and I’m very smart.” So it’s a wide-ranging group of people all over the world, from many different occupations. One of the things people like about the OEIS is this opportunity to collaborate, to exchange emails with professionals. It’s one of the few opportunities that most people have to talk to a real mathematician.

Do you feel that there is a divide between “serious mathematics” and “recreational mathematics”? Or do you tend not to think in those terms?

I don’t think in those terms. I don’t think there’s much difference. If you look hard enough, you can find interesting mathematics anywhere.

This article was reprinted on Wired.com.

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  • Fabulous piece!… I suspect a lot of folks (like me) are aware of the OEIS, yet found MANY of the above details fresh and wonderful! Neil has created a treasure chest…

  • In my experience, it's often called "Sloane's" by users, not "Sloane." (But maybe there are regional dialects.) Either way, it's a great resource. Thanks for the article.

  • Yes, here in Washington it is called Sloane's.
    The central feature of the repository is that if you're stuck trying to compute some mysterious sequence, you can just type in the first 5 terms or so, that maybe you can work out by hand. Then if you're lucky Sloane's will give you a formula for the whole thing, or at least some references to related information. It is impossible to google this sort of thing, because you don't know the name that someone else has given it.

  • I am both a user and sometime contributor. Sloane warrants the praise, a very useful invention, and I believe it will have greater future, lasting impact.

  • The OEIS has an epic importance to the Math. I agree with some when they say that Sloane is one of the most important mathematicians of this time.

  • Thanks to Erica Klarreich for a really excellent article. Although it is indeed often referred to as "Sloane", it should be mentioned that since 2009 the intellectual property of the OEIS has been owned by a 501(c)(3) charity, the OEIS Foundation Inc. If the article had been longer I would have liked to have mentioned the names of some of the brilliant, hardworking, diligent, unpaid editors who keep the OEIS running on a day-to-day basis, for example David Applegate, Joerg Arndt, Russ Cox, Harvey Dale, Olivier Gerard, Charles Greathouse, Maximilian Hasler, Alois Heinz, Robert Israel, Michel Marcus, Richard Mathar, Jon Schoenfield (the list could easily be twice as long). One favorite sequence that didn't get mentioned is the Peaceable Queens sequence (place equal numbers of black and white queens on an nXn board so that no queen attacks a queen of the other color – these are peaceable queens!). This sequence won the competition to see which entry would become sequence A250000. I like to think it illustrates the OEIS's modest contribution towards international cooperation and world peace. Finally, a correction that didn't make it into the article: when I moved the database to the Internet in 1996, this was done with the help of Simon Plouffe.

  • I found the OEIS several years back while researching magic numbers and atomic nuclear shell structure looking for mathematical patterning. The site was very useful, and I joined their discussion list, even interacting with Neil several times. Everyone was very friendly and informative. I ended up contributing new details for several of the existing sequences relevant to my study, as well as one or two new sequences.

  • Perhaps I don't understand but in the example given in the article in which circles are arranged on a plane, the explanation says that only two can cross at any given point. The image of the three black circles on the left shows three circles crossing at a single point. Did I misunderstand the rule or is this image an error?

  • I was doing some sequencing of Pythagorean Triangles and found two of the three sequences for the sides had been cataloged by Sloane's in an unrelated work (Pell's). I then used these sequences and the seeming unrelated work and found a relationship did exist between these areas. Great resource which will undoubted lead to many more such accidental discoveries.

  • In regard to the final question posed by
    Erica Klarreich to Neil J. A. Sloane, perhaps
    there is an area of overlap, to be christened
    "serious recreational mathematics", in
    the "Fun and Games" curiosity department?

    (In my view, crossword puzzles are not
    SRM, because they rely too much on semantics
    and orthography; neither are many contrived logic
    puzzles SRM, because they often comprise
    straightforward, though complex applications
    of logic specific to their case.)

    —- —- —- —-
    In the 06 August 2015 Quanta Magazine Article by Erica Klarreich on Neil Sloane


    Klarreich (Quanta):
    Do you feel that there is a divide between “serious mathematics” and
    “recreational mathematics”? Or do you tend not to think in those terms?
    I don’t think in those terms. I don’t think there’s much difference. If you
    look hard enough, you can find interesting mathematics anywhere.
    —- —- —- —-

  • Frank McCullar is correct that one of the drawings appears to show three circles meeting at a point, but this is my fault for making the circles too thick. In fact there is a small empty space where the three circles would have met. For a better illustration, showing all 14 arrangements, see https://oeis.org/A250001

  • This work is amazing. Some scientists work hard for a Nobel price, and most have their names forgotten after a few generations. What Neil did will be preciously kept and upgraded for generations to come (one might wonder why its data is not yet integrated with Google search, but truth has become Google is now lagging in creativity and fore-front technologies).

    Future generations will remember Neil's name and work.

  • At one point in Math, as an undergraduate, Series were part of the curiculum. It was fun to devise or decode them. Given our mischeviousness we devised a new field, Sequences. These were a string of numbers, or other symbols, for which there was a rule, but not necessarily a Mathematical one. An early, and trivial, example was o, t, t, f, f, s, s, e, The objective was to find the next sympol after e . If it is not clear then count them out loud. More generally, isn't this a code breaker's dilema?

  • I have discovered oeis by internet
    So I have wrote to nja sloane by mail to present some calculs about number's sequences-
    after being user I was turned contributor of some sequences.
    I have found this article on SLOANE because I am seaching the article of Erika about
    the 2 mathematicians of stanford(prime conspiracy) very interisting showing that prime –
    numbers are not aleatories-it's sure by the definition- I am not surprised but estonished that since Hardy and Littlewood( 90 years ago) nobody makes statistics on primes apparition

  • I found OEIS around 2006, and was immensely happy for it. Since then used it, and even contributed some sequences. One of my deepest questions in maths, leading to a short project, was designing a tool that will find possible sources for given numbers or sequences –
    CONTEXYPHER: a research project for identifying the source and precise formulation of given unknown numbers, such as CubeRoot(2PI-6) , (exp(2/3)+1)/(exp(4/3)-1), sqrt(1492)-38. It involves Progressive-Continuous-Fractions, among other difficulties, such as the identification of a third-degree equation with integer terms (given a real root).
    After discovering that this project is being thoroughly implemented, it is largely abandoned. Inverse Symbolic Calculator (http://isc.carma.newcastle.edu.au/index)
    Highly recommended to all "serious" or "amateur" mathematicians, seeking more context, although it isn't fully functional, and probably undervalued. But this tool holds a major key to future maths.

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