I noticed something a couple days ago – I’m 32 now, and it’s 2016, and 32 is a factor of .
Last year, though, had the same property – 31 is a factor of .
So in which years will my age on New Year’s Day, divide the current year ? These are just the factors of .
(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,
- it’s a triangular number,
- it’s the number of ways to arrange two same-color pawns on a chessboard,
- 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 , 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 and inclusive.