Interesting Numbers
About an year ago, my Ph.D. advisor (Douglas Hofstadter) showed me a graph he had been drawing by hand. The x-axis consisted of the natural numbers: 1, 2, 3… There was a stack of dots above each, forming the y-axis. There was a red dot above all the squares (1, 4, 9…), a blue dot over the triangular numbers (1, 3, 6 …), a yellow dot over powers of “2″ (1, 2, 4, 8, 16…) and so forth, the idea being that the more interesting a number is, the higher the stack of dots above it.
I decided to help and automate the process. I downloaded the Encyclopedia of Integer Sequences, and created a different graph, based on the same idea. The x-axis was the same. Instead of dots of various colors, I just drew the regular, everyday graph, with the y value corresponding to an integer being how many distinct sequences in the Encyclopedia it was present in.
I had access to a whopping 128000 sequences. Two graphs are shown below. One shows numbers upto two hundred, the other between 4000 and 4200.
I can also provide a file containing the “interestingness values” of numbers upto 50,000, if you like.
The first five numbers that are “boring” are, incidentally, 8795, 9734, 9935, 10017, 10418.
March 8, 2008 at 12:33 am
So what happens between 4091 and 4098?
Does that spike repeat itself often?
March 8, 2008 at 12:53 am
You get 4096 (a power of 2, a cube, smallest number with 13 factors. It is a Jordan-Polya number (whatever that is), it is also in the expansion of sin(x)*exp(x), and so forth).
See http://www.research.att.com/~njas/sequences/?q=4096
There are a few spikes, here and there, but the encyclopedia is biased towards smaller numbers in the sense that only the first few hundred or less terms of sequences are shown. Thus, I believe that spikes (in the data from the encyclopedia) get artificially sparser. But I have not looked very carefully.
March 9, 2008 at 9:24 am
Seems to me there’s an element of self-correction here — the sequence of “boring numbers” is itself inherently interesting, isn’t it?