Sloane's Gap

Figure 1. The number of entries into the OEIS for each number between 1-10,000. 

The Online Encyclopaedia of Integer Sequences (OEIS) is a database that lists the first few numbers in various whole number sequences. For example, multiples of 3 or square numbers. Each number has a value for the frequency that it occurs in the database e.g. the number 1729 occurs in the database 653 times. 

Figure 1 displays a distribution of the number of occurrences that a number has in the database. There are two clear groups: group 0 and group 1. Group 0 consists of numbers with fewer entries in the database than group 1. Group 1 has some interesting properties. It consists mostly of prime numbers and squares. In fact, 70% of the numbers in group 1 being either a square or prime. This interested mathematicians, could this effect be explained mathematically?

Gauvrit, Delahaye and Zenil (2011) were interested in explaining this gap, with a particularly emphasis on mathematical complexity being the driving force. The reasoning behind this is that simple sequences can be broken up in multiple ways. For example, square numbers form a simple sequence and can be broken into more sequences very simply, e.g. squared odd numbers or squared even numbers. However, this explanation proves inadequate as it can explain some variation in the numbers but does not predict a gap.

The researchers instead concluded that there was some interaction between the simplicity of numbers and the mathematicians recording sequences into the OEIS. The driving force for this effect may be the availability heuristic, which states that the ease of recall is used as an indicator of importance (Tversky & Kahneman, 1973). This may suggest that when mathematicians are creating new sequences, their starting point may be the square numbers or primes and that leads to these numbers having a higher number of entries in the OEIS.