Frequency of appearance in the OEIS database

In 2008 Ph. Guglielmetti (aka “Dr. Goulu”) had the strange idea of searching the “least interesting [positive] integer” in OEIS, i.e the smallest positive integer which wasn’t referenced in any sequence, at least not in the first terms shown or used for searching. Of course they all belong at least to A000027.

At the time of writing,

• 8795, 9935, 11147, 11446, 11612, 11630 were the first integers not appearing in any OEIS sequence,
• 8267 and 9734 were the first integers lower than 10000 appearing only once,
• 7495, 8758 and 9820 appeared in 2 sequences.

As Neil Sloane didn’t consider this subject very interesting, Ph. Guglielmetti developed a collaboration with Jean-Paul Delahaye who suggested to make a plot of the occurrences, which revealed a strange “gap” in the graph of the “interestingness” of numbers measured by the number of their occurrences.

Plot of the occurrences, revealing a strange “gap” in the graph of the “interestingness” of numbers measured by the number of their occurences.

(...) les nombres entiers se divisent assez clairement en 2 groupes : les nombres “intéressants” et les nombres “inintéressants”. Il y a étonnament peu de nombres moyennement intéressants. L’article de Delahaye se termine par un appel à expliquer ce phénomène. (...)

which could be translated as

(...) the integers are quite clearly divided in 2 groups : the “interesting” numbers and the “uninteresting” numbers. There are surprisingly few “more or less interesting” numbers. The Delahaye article ends by a asking for an explanation of this phenomenon. (...)

This led to further research and publications by Gauvrit and Delahaye on what they called “Sloane’s gap.”

(...) The graphic representation of the frequency with which a number

n

as a function of

n

appears in that database shows that the underlying function decreases fast, and that the points are distributed in a cloud, seemingly split into two by a clear zone that will be referred to here as “Sloane’s Gap.” (...)