11630 is the First Uninteresting Number
There’s an old math paradox that says that all natural numbers are interesting, since otherwise there would have to be a smallest uninteresting number, and that in itself is pretty interesting. Of course, this is meant to show that ideas in the English language do not always translate to well-defined mathematical concepts, but let’s ignore our better mathematical sense and tinker with the idea of how interesting different numbers are a little bit. In particular, I claim that 11630 is utterly bland and uninteresting.
Before saying why 11630 is uninteresting, I should probably say what I consider “interesting” to even mean. Interesting, to me, means that it has some (semi-unique) mathematical property that sets it apart from other numbers. 11 is interesting because it is prime, 16 is interesting because it is a perfect power (16 = 42), and so on. Clearly, there is some ambiguity in this definition, since one could consider composite numbers interesting, just as I considered prime numbers interesting. Additionally, do we consider 2719 interesting simply because it is prime? I’d say no, since there are hundreds of prime numbers that come before it — perhaps only the first few numbers that satisfy a given property should be considered interesting as a result of it?
Using these ideas, it seems like determining how interesting a number is would be a task perfectly suited to the Online Encyclopedia of Integer sequences (OEIS). If you’re unfamiliar with it (i.e., if you’re not a math person and have no place reading my blog), the OEIS is a database containing thousands of (you guessed it) integer sequences that have been submitted by users over the last decade or so (such as the sequence of prime numbers 2, 3, 5, 7, 11,… and the sequence of perfect powers 1, 4, 8, 9, 16,…). Presumably, if an integer is interesting then it will appear in at least one or two of the 159437 sequences contained in the database, right? Indeed, it seems that we can get a rough idea of how interesting a number is by looking at how many sequences that number appears in in the database compared to other numbers of similar size.
11630 is the first number that is not listed in a single sequence in the OEIS. It is not prime, nor is it highly composite (11630 = 2×5×1163). It doesn’t have any particularly notable residue properties, and it doesn’t come up in counting problems. It’s boring in every way, and it seems as though not a single mathematician has found a use for it in the last dozen or so years (let me know if you’ve discovered otherwise).
What Numbers are Interesting?
First off, I’m not going to deal with particularly small numbers (say in the range of 1 – 50) since, as the strong law of small numbers quips, these numbers will appear all over the place just because they’re small. You could probably argue that most (if not all) of them are interesting, so I’ll instead take a look at a couple larger numbers that are particularly interesting.
The number 421 appears in some 1894 sequences, while most numbers that size appear in about 940 sequences. This seems to indicate that 421 is a particularly interesting number, but why? What’s so special about 421? Well, it’s prime (in fact, it’s a twin prime, Pythagorean prime, cuban prime, lucky number of Euler prime, additive prime, and irregular prime), it’s congruent to 1 mod 2,3,4, 5, 6,7, 10, 12, it’s the sum of five primes, and 4212 = 4202 + 292. Similarly, 512 appears in 2116 sequences even though most numbers around 512 appear in about 800 sequences. This is perhaps less surprising than 421, since 512 = 29 = 83 = 162 + 162 is a number that somehow seems “nice” due to it being a perfect power. Additionally, 512 is a Leyland number, Harshad number, and it comes up in all sorts of counting problems.
What of the Paradox?
Recalling the paradox from earlier, we are now forced to ask ourselves whether or not 11630 is now interesting as a result of it being the first number not included in the OEIS. Rather than come up with an answer, I’m going to take the easy way out and let the OEIS decide. The sequence of uninteresting numbers is 11630, 12067, 12407, 12887, 13258, 13794, 13882, 13982, 14018, 14163,… Let’s submit that to the OEIS and see if they consider it to be interesting or not.
Update [June 13, 2009]: I got word back via e-mail today that this sequence didn’t make the cut. So there you have it — these numbers truly are uninteresting.
Update [October 7, 2011]: Interested readers might want to check out this paper, which explores similar questions and mentions the numbers computed here.
Update [November 22, 2013]: A whole bunch of these numbers have been added to the OEIS lately, making 14228 the new first uninteresting number.