# 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.