SLOANE'S GAP

*Editors Note: The following submission by*

**Emil H. ('15)**explores a distribution of the integer sequences recorded on their website and the frequency with which they occur. You can find almost anything online. A particularly interesting website to visit would be oeis.com, or The Online Encyclopedia of Integer Sequences. This website was introduced to me by a Numberphile YouTube video[1] about a paper written by three Mathematicians Nicolas Gauvrit, Jean-Paul Delahaye, and Hector Zenil[2]. At first encounter, the OEIS is mind-numbingly huge. As of March 13, 2014, 6:51 Pm it contains 238,882 integer sequences. When I first looked at the website I was baffled by the scale of what initially seems to be a massive database of lists of numbers. However, describing it as just a collection of lists of numbers totally undermines its significance. While it started as a way to catalog various sequences, the OEIS has now turned into a dictionary of numbers where anyone can search a particular integer and find out many of its properties. For example searching the number 2 would tell you that it is the second number in the series x=2n-1 or the third number in the series x=1+n where n is the term number as well as 214,789 other (probably more interesting) results. Additionally, anyone can submit a sequence and it will get posted after approval by the site’s editorial committee. This makes for a large, diverse group of sequences that comprise the encyclopedia. All this makes the OEIS a great place to research specific integers, as well as analyze certain trends that may or may not be revealing about some possible biases the seemingly objective math world may have.

The Numberphile video on YouTube I mentioned was about a particularly interesting phenomenon that occurs in the OEIS. Three Mathematicians who authored the paper the Numberphile video is about decided to make a graph that would represent this encyclopedia. On the x axis there are integers n, and the number of times that integer appears in the OEIS is represented as N(n) on the y-axis. What they found, in my humble opinion, is quite astounding.

The Numberphile video on YouTube I mentioned was about a particularly interesting phenomenon that occurs in the OEIS. Three Mathematicians who authored the paper the Numberphile video is about decided to make a graph that would represent this encyclopedia. On the x axis there are integers n, and the number of times that integer appears in the OEIS is represented as N(n) on the y-axis. What they found, in my humble opinion, is quite astounding.

This is the graph they came up with. Not totally surprisingly, numbers 1,2,3 and other basic counting numbers appear very frequently, and larger numbers like 7000 don’t appear as many times. However the focus of these three mathematicians’ paper is the gap that appears between the two clouds of points. This space mostly void of points is called Sloane’s Gap. The gap is named after Neil Sloane who created OEIS in 1996 (it wasn’t digital then) and still works on it to this day. This result was very unexpected and baffling to the three mathematicians. Is this gap caused by the properties of sequences themselves, or is there something else at play here? The three mathematicians decided to create a randomly generated list of sequences with 8 million total points as a way to test what a non-human generated graph might look like. This is what their computer-generated integer sequences looked like:

This is very illuminating. Something about a human-generated database of sequences produces a certain pattern that doesn’t appear on an arguably less biased computer generated bunch of sequences. What is causing the numbers in the top band of the human-generated graph to be so much more popular than the numbers in the bottom band? Why is this mysterious gap appearing?

Thanks to the OEIS’s ability to let us find out certain properties of specific integers, we can actually find out some of the properties of the numbers in the top band (group 1) as compared to those of the bottom band (group 2). There are some very specific reasons for why the numbers in the top band are more frequently appearing than those in the less popular bottom band. It is important to note that the three mathematicians in their paper chose to only analyze trends in the graph for numbers greater than 301, because before 301 the gap becomes very small and less clear. With that said, the trends from numbers 301-10000 are very interesting. 95% of square numbers appear in the top band, 99.7% of prime numbers appear in the top band, as well as the majority of powers of two. All numbers with 10 or more prime factors appear in the top band. Less than 10% of the numbers in the top band have no discernible properties. All the other numbers without any particularly interesting properties fall into the bottom band.

In conclusion, this trend for numbers with certain properties to be more represented in the OEIS is exemplary of a bias mathematicians may have of favoring these numbers over others. In their quest to find neat things out about primes, squares, and other numbers with desirable, interesting properties, the mathematicians submitting to the OEIS have focused their efforts on these types of numbers, while neglecting others. The three mathematicians who published this information believe that Sloane’s gap is caused by a human interest in numbers with specific qualities, and I agree with them. This graph shows us humans that we spend a disproportionate amount of our time around certain numbers. This is the intersection between human curiosity and numbers. Does this mean we should stop neglecting non-primes and non-squares and give other numbers some attention? Not necessarily, but it’s important to recognize our natural biases and inclinations if we want to study sequences in a more objective way.

In conclusion, this trend for numbers with certain properties to be more represented in the OEIS is exemplary of a bias mathematicians may have of favoring these numbers over others. In their quest to find neat things out about primes, squares, and other numbers with desirable, interesting properties, the mathematicians submitting to the OEIS have focused their efforts on these types of numbers, while neglecting others. The three mathematicians who published this information believe that Sloane’s gap is caused by a human interest in numbers with specific qualities, and I agree with them. This graph shows us humans that we spend a disproportionate amount of our time around certain numbers. This is the intersection between human curiosity and numbers. Does this mean we should stop neglecting non-primes and non-squares and give other numbers some attention? Not necessarily, but it’s important to recognize our natural biases and inclinations if we want to study sequences in a more objective way.