# Smallest uninteresting sequence?

In a course I have been asked to play with the Online Encyclopedia of Integer Sequences (OEIS, oeis.org).

OK, so I shot in my birth month, day, and year (divided into two two-digit numbers), just to see what I would bring up. And the answer was, nothing. I was a little surprised (especially since my birth year is 64, which is, I figure, a good number for finding a sequence, being a square and a cube and a power of two and all that). And my sequence of four numbers was strictly ascending.

So I remembered some nonsense about the first uninteresting number (was it Hardy and Ramanujan playing? Silly, right, if 1 is interesting, and 2 is interesting, and … once you get to a number that is NOT interesting, well, that fact makes it interesting).

So, can we do the same thing, play the same nonsense for a sequence? What is the smallest sequence not in the OEIS (and can we use that fact to wrangle a place for that sequence, and a real citation, in the encyclopedia?)

So let’s look for the smallest sequence of four terms (ascending) that’s not in there. Why 4? Because that’s where I started. Why ascending? Because there have to be some arbitrary rules… and this one is useful.

Certainly we have a problem with the idea of “small”. We could defined the size of the sequence to be the highest number in the sequence (like comparing junk poker hands), so 2, 10, 11, 13 (not in there), is smaller than 2, 10, 11, 14 (not in there) because we begin by comparing the 13 and the 14… Or should we look at the sum of the 4 terms, in which case 3, 4, 5, 19 (not in there) is better? Or the sum of the squares…

And once we decide what small is, we would still need to find it…

By the way, 1,2,3,n is not in there, for what smallest value of n?

Nothing for 1, 2, 3, 51, but I only searched through some of the smaller ones.

Close – 1,2,3,45 also turns up nothing.

But if we alter what smallest means… I have been working with where is as small as possible. I have also been thinking about a sequence with the smallest possible sum… I think those are more interesting. 45 is pretty big, and there are candidates with sums around half of 51…

I like 2, 10, 11, 13

For an uninteresting infinite sequence of strictly increasing positive integers, a_n = n is smaller, but so are uncountable many such uninteresting sequences. There is no least, but their liminf is a_n = n.