by What is your favorite number? Many people may have an irrational number in mind, such as pi, Euler’s number (m) or the square root of 2. But even among natural numbers, you can find values encountered in a wide variety of settings: the seven dwarfs, the seven deadly sins, 13 as an unlucky number — and 42, which became popular from the novel The Hitchhiker’s Guide to the Galaxy by Douglas Adams.
What about a larger value like 1.729? The number certainly doesn’t seem particularly exciting to most. At first glance, it seems to be completely boring. After all, it is neither a prime number nor a power of 2 nor a square number. The numbers don’t follow any obvious pattern either. That’s what the mathematician Godfrey Harold Hardy (1877–1947) thought when he got into a taxi with the identification number 1729. At the time, he was visiting his sick colleague Srinivasa Ramanujan (1887–1920) in hospital and told him about the “boring” number cabin. He hoped it wasn’t a bad omen. Ramanujan immediately contradicted his friend: “It is a very interesting number. is the smallest number that can be expressed as the sum of two cubes in two different ways’.
Now you might be wondering if there might be some number that isn’t interesting at all. This question quickly leads to a paradox: whether there really is a value n that has no exciting qualities, then that very fact makes it special. But there is indeed a way to determine the interesting properties of a number in a fairly objective way—and much to the surprise of mathematicians, 2009 research suggested that natural numbers (positive integers) fall into two clearly defined camps: exciting and boring values.
A comprehensive encyclopedia of number sequences provides a means of exploring these two contrasting categories. Mathematician Neil Sloane came up with the idea for such a collection in 1963, when he was writing his PhD thesis. At that time, he had to calculate the height of the values in a type of graph called a tree network and he came across a sequence of numbers: 0, 1, 8, 78, 944,… He didn’t yet know how to calculate the numbers in this sequence exactly and he would like to know if his colleagues had already encountered a similar series during their research. But unlike logarithms or formulas, there was no register for sequences of numbers. And so, 10 years later, Sloane published his first encyclopedia, A Handbook of Integer Sequences, which contained about 2,400 sequences that also proved useful for performing some calculations. The book met with enormous acceptance: “There is the Old Testament, the New Testament and the Handbook of Integer Sequences,” wrote one enthusiastic reader, according to Sloane.
In the years that followed, numerous submissions with more sequences reached Sloane, and scientific papers with new sequences of numbers also appeared. In 1995 this prompted the mathematician, together with his colleague Simon Plouffe, to publish The Encyclopedia of Integer Sequences , which contained approximately 5,500 sequences. The content continued to grow unabated, but the Internet made it possible to control the flood of data: in 1996, the Electronic Encyclopedia of Integer Sequences (OEIS) appeared in a format unconstrained by limitations on the number of sequences that could be recorded. As of March 2023, it contains just over 360,000 entries. Submissions can be made by anyone: a person making an entry need only explain how the sequence was created and why it is interesting, as well as provide examples that explain the first terms. Reviewers then check the entry and publish it if it meets these criteria.
In addition to well-known sequences such as prime numbers (2, 3, 5, 7, 11,…), powers of 2 (2, 4, 8, 16, 32,…) or the Fibonacci sequence (1 , 1, 2, 3, 5, 8, 13,…), the OEIS catalog also contains exotic examples such as the number of ways to construct a fixed tower from n Lego blocks with two by four studs, (1, 24, 1,560, 119,580, 10,166,403,…) or the “lazy caterer sequence” (1, 2, 4, 7, 11, 16, 22, 29,… ) , the maximum number of pie pieces that can be achieved with n cuts.
Because about 130 people review the submitted number sequences, and because the list of these obvious candidates has been around for several decades and is quite well known in the mathematical community, the collection is intended to be an objective selection of all sequences. This makes the OEIS catalog suitable for studying the popularity of numbers. Accordingly, the more often a number appears in the list, the more interesting it is.
At least, that was the thinking of Philippe Guglielmetti, who runs the French-language blog Dr. Goulu. In one post, Guglielmetti recalled a former math teacher’s claim that 1,548 was an arbitrary number with no special property. This number actually appears 326 times in the OEIS directory. An example: it appears as ‘final period of a single cell in the rule 110 cellular automaton in a circular universe of width n.” Hardy was also wrong to call the cabin number 1729 boring: 1,729 appears 918 times in the database (and also often in the TV show Futurama).
So Guglielmetti was looking for really boring numbers: ones that barely appear in the OEIS catalog. The latter occurs, for example, with the number 20,067. As of March, it is the smallest number that does not appear in any of the many stored number sequences. (This is precisely because the database only stores the first 180 or so characters of a sequence of numbers, however – otherwise, every number would appear in the OEIS list of positive integers.) So the value 20,067 looks pretty boring. In contrast, there are six entries for the number 20,068, which follows it.
But there is no universal law of boring numbers, and the situation of 20,067 can change. Perhaps during the writing of this article, a new sequence was discovered in which 20,067 occur between the first 180 characters. However, OEIS records for a given number are suitable as a measure of how interesting that number is.
Guglielmetti proceeded to count all the entries extracted in order for the natural numbers and plotted the result. He found a cloud of points in the form of a broad curve that slopes toward large values. This is not surprising since only the first members of a sequence are stored in the OEIS directory. What is surprising, however, is that the curve consists of two zones separated by a clearly visible gap. Thus, a natural number appears either extremely often or extremely rarely in the OEIS database.
Fascinated by this result, Guglielmetti turned to mathematician Jean-Paul Delahaye, who regularly writes popular science articles on For Science, Scientific American‘s French-language sister publication. He wanted to know if experts had already studied this phenomenon. This did not happen, so Delahaye took up the matter with his colleagues Nicolas Gauvrit and Hector Zenil and investigated it more closely. They used results from algorithmic information theory, which measures the complexity of an expression by the length of the shortest algorithm that describes the expression. For example, an arbitrary five-digit number such as 47,934 is more difficult to describe (“the sequence of digits 4, 7, 9, 3, 4”) than 16,384 (214). According to a theorem from information theory, numbers with many properties usually also have low complexity. That is, values that appear frequently in the OEIS catalog are more likely to be simple in description. Delahaye, Gauvrit, and Zenil were able to show that information theory predicts a similar trajectory for the complexity of the natural numbers as seen in Guglielmetti’s curve. But that doesn’t explain the gaping hole in this curve, known as ‘Sloane’s gap’, by Neil Sloane.
The three mathematicians suggested that the gap arises from social factors such as a preference for certain numbers. To document this, they ran what’s known as a Monte Carlo simulation: they designed a function that maps natural numbers to natural numbers—and does so in such a way that small numbers come out more often than large ones. The researchers put random values into the function and plotted the results according to their frequency. This produced a fuzzy, sloping curve similar to that of the data in the OEIS catalogue. And as in information theory analysis, there is no trace of a vacuum.
To better understand how the gap occurs, we need to see which numbers fall into which band. For small values up to about 300, the Sloane gap is not very sharp. Only for larger numbers does the gap widen significantly: about 18 percent of all numbers between 300 and 10,000 are in the “interesting” zone, while the remaining 82 percent fall into the “boring” range. As it turns out, the band of interest includes about 95.2 percent of all square numbers and 99.7 percent of prime numbers, as well as 39 percent of numbers with many prime factors. These three categories already account for nearly 88 percent of the interesting band. The remaining values have impressive properties such as 1111 or the types 2n + 1 and 2n – 1, respectively.
According to information theory, the numbers that should be of particular interest are those that have low complexity, that is, are easy to express. But if mathematicians find some values more exciting than others of equal complexity, this can lead to Sloane’s gap, as Delahaye, Gauvrit, and Zenil argue. For example: 2n + 1 and 2n + 2 is equally complex from an information theory point of view, but only values of the first type are in the “zone of interest”. This is because such numbers enable the study of prime numbers, which is why they appear in many different contexts.
So the breakdown into interesting and boring numbers seems to come from judgments we make, such as assigning importance to prime numbers. If you want to give a really creative answer when asked what your favorite number is, you could come up with a number like 20,067, which doesn’t yet have an entry in Sloane’s encyclopedia.
This article originally appeared on Spektrum der Wissenschaft and reproduced with permission.
