The On-Line Encyclopedia of Integer Sequences® (OEIS) is as its name describes. You can consult the encyclopedia to see if a pattern of numbers has entered humankind’s online knowledge store. For example, A000037 describes the sequence of numbers that are not squares: 2, 3, 5, 6 and so on. Or let’s say this one time you sat at your desk to type something, and while you typed, your cat put his paw on the ‘1’ key. At first you did not notice because all you wanted to do was insert three new lines on your document – then you looked at the screen:
Surprisingly, looking up this pattern in the OEIS reveals that these are primes of the form (10^n – 1)/9. Your cat is a(n annoying) genius.
In my opinion, the OEIS is best used as a brain trainer. I say this because people in Australia (as in other developed nations) are tending to live longer, and so are more prone to developing age-related brain diseases such as Alzheimer’s. As the brain ages, its ability to re-wire itself diminishes proportional to how much it is being challenged. This is sometimes called the use it or lose it principle; a main tenet in the theory of brain plasticity. In other words, if you stop making a cognitive effort later in life, your brain will cease to form new interconnections. It is a bit like the Australian roads to recovery program. Local roads that are used a lot receive federal priority funding for extensions, so that people can travel between areas more quickly.
If you already do some form of cognitive muscle training, like the daily crossword or Sudoku, then sudo-kudos to you (see what I did there?). In a recent study, taking up crossword puzzles later in life is associated with a delay in memory decline of more than two years. Moreover, another study has shown a consistent link between self-reported cognitive activity, and (a) grey-matter volume in the hippocampus and other structural correlates of memory; (b) better performance in memory, verbal learning, and cognitive speed and flexibility tests.
Even if you are a staunch crossword enthusiast, you may wish to let your dendrites branch out to the fabulous world of recreational math. A large-n study has found that while crosswords are good for cognitive function, they are no match for a comprehensive brain training program. Granted, the study was funded by the creators of the ever-popular Lumosity program, but the use it or lose it principle holds water – if crosswords become too familiar, they will cease to improve cognitive function beyond a certain degree. Besides, you may find that a new activity, like solving equations for leisure, inspires new approaches to more familiar problems.
The OEIS is not your typical encyclopedia, in that you can manipulate its entries in some surprising ways. For instance, you can convert your favourite integer sequence into music, or draw curves that express the relationship between two sequences. The OEIS even has a dedicated puzzles section. If you are feeling especially brave, you can subject a new sequence to peer-review. So far, by toying around with numbers and functions in MS Excel (one of my favourite programs), I have published three sequences in the OEIS. I have become so infatuated with discovering new integer sequences, that an accomplished, career mathematician wrote me the following:
Justin, your submissions are consuming a great deal of the editors’ time. Maybe take a break from submitting anything else, OK?
I will not reveal which three integer sequences I am credited as discovering, but for fun, below are the first 10 terms of each, starting at n = 0. Without consulting the OEIS, see if you can guess what the next number in each sequence will be. Can you work out a formula that is the throughput linking each input term n to its output term a(n)?
- 0, 1, 3, 12, 30, 120, 315, 1344, 3780, 17280, ????
- 0, 9, 30, 60, 120, 189, 432, 630, 825, 1122, ????
- 0, 6, 18, 75, 126, 288, 405, 726, 936, 1470, ????
Bonus question: Starting with n = 2, if you plot the ratio of each term to its preceding term in log-log space, which of the above three sequences results in the following pattern?
While I do not expect you to answer the esoteric bonus question, I have included the graph because I made it in Excel and I think it looks rad. It reminds me of the flaming tracks left behind by the DeLorean time machine, shortly after the magic velocity point of 88 mi/h has been reached.
No matter your current arithmetic skill or confidence level, I do recommend that you consider crunching some numbers toward better brain health. When it comes to mathematics, I consider myself to be amateur at best, but that is beside the point. Whether playing the violin (like my daughter), or building in Minecraft (like my son), you can do things for the hell of it and still reap the cognitive benefits, so long as you are open to challenging yourself.
As a mere number hobbyist, I asked an accomplished, career mathematician: “Is it possible to continue contributing even small amounts of knowledge to the world of numbers and mathematics, without formal training?” I will leave you with the mathematician’s response:
Well, math is a bit like the violin – it is hard to make any real contribution without many years of training. On the other hand, you can get a great deal of pleasure from reading the great books. Tom Korner’s The Pleasures of Counting is one of the greatest. Or Hardy and Wright’s The Theory of Numbers. Or Concrete Math by Graham et al.