# Some properties of 2016

Happy 2016!

I noticed something a couple days ago – I’m 32 now, and it’s 2016, and 32 is a factor of $2016 = 2^5 \times 3^2 \times 7$.

Last year, though, had the same property – 31 is a factor of $2015 = 5 \times 13 \times 31$.

So in which years will my age on New Year’s Day, $x$ divide the current year $1984 + x$?  These are just the factors of $1984 = 2^6 \times 31$.

(This seems like a nice problem – which numbers have two consecutive factors?  But in fact that’s just a characterization of the even numbers.)

Some other properties of 2016:

• it has a nice binary representation as 11111100000; alternatively it’s the sum of consecutive powers of 2, $2^{11} + 2^{10} + 2^9 + 2^8 + 2^7 + 2^6$
• it’s a triangular number, $1 + 2 + 3 + \cdots + 63 = (64 \times 63)/2$
• it’s the number of ways to arrange two same-color pawns on a chessboard, ${8^2 \choose 2}$
• it’s divisible by every single-digit number except 5

2016 appears to be a relatively interesting number, appearing in 784 sequences in the OEIS – compare 179 sequences containing 2015 and 428 containing 2017.  This is probably traceable to the fact that it has many prime factors – all of the above are essentially divisibility properties – and this is the big idea in the paper on “Sloane’s gap” by Nicolas Gauvrit, Jean-Paul Delahaye and Hector Zenil.    (Don’t like reading?  Check out the Numberphile video instead.)

In fact, back in 2009 Philippe Guglielmetti put together a spreadsheet giving the number of appearances in OEIS of each number up to $2^{16}$, which you can download from this post (click on “feuille Excel”); at the time 2016 appeared in the OEIS more often than any other number between $1729 = 12^3 + 1$ and $2047 = 2^{12}-1$ inclusive.

## 3 thoughts on “Some properties of 2016”

1. I’m pleased to see that if I count any moment at which my age divides the year, not just the ones that occur on January 1st, those ages are roughly evenly distributed throughout the rest of my life: ages 28, 39, 51, and 71. That’s a little more special than being in one of the primes of my life, but not so special as being a perfect age—I’ll likely get only one more chance at that!

2. 2016 is the order of GL(2,7) – the invertible matrices over GF(7).
That’s pretty special!