This will never happen again…

Here’s a quick, fun tidbit about dates. I was thinking about the factorization of 2013 and the next few years, and I noticed something cool. (Note this is good for psyching out math contests, which often have problems with the current year in them.)

2013 = 3 * 11 * 61

2014 = 2 * 19 * 53

2015 = 5 * 13 * 31

Each date factors into primes as (single digit) * (teen) * (double digit bigger than teen). (I’m counting 11 as a teen, i.e. I really mean a prime between 11 and 19.) So I wondered how often you get three such numbers in a row, and I wrote a little Matlab code. It turns out that these three are the only example! In fact there are only three other consecutive pairs: 1221,1222; 1598,1599; and 2821,2822.

I can’t imagine that this kind of factorization has any real importance, but I thought it was cool. The next year, 2016, is quite different, but also cool: it is

2016 = 32 * 63 = 2^5 * 3^2 * 7

which is (power of two) * (power of two – 1), and which is also “7-smooth”, meaning that it has no prime factor larger than 7. (These are sometimes called “highly composite”, but that has a different, and more interesting, meaning, about which I made some videos last year.) The 7-smooth numbers are pretty sparse; the only ones occuring in anyone’s lifetime who is alive right now are 1920, 1944, 1960, 2000, 2016, 2025, 2048, 2058, 2100. Note that 2048 = 2^11, which will be cause for worldwide, year-long celebration, no doubt. 🙂

Speaking of numbers of the form (power of two) * (power of two – 1), if a number n is n = 2^(k-1) * (2^k – 1), and (2^k – 1) is prime (a Mersenne prime), then n is a perfect number. Now 2016 doesn’t fall into that category, since 63 is not prime, but it is a number with a particularly large sum of all of its factors, called sigma(n) (which, for a perfect number n, would equal exactly 2n). The sum of factors of 2016 is sigma(2016) = 6552, which is higher than for any other number in the years 1900-2099 except for 1980 (which ties it with sigma(1980) = 6552). The year 2040 is very close, with sigma(2040) = 6480, and 2100 beats it with sigma(2100) = 6944.

The story is very similar if you just look at the number of divisors (often called sigma_0(n) or tau(n)). The number n = 2016 again beats everything except 1980 and 2100, which both tie it with tau(n) = 36. So in 2016, let’s make sure to do a lot of division!

If you watch my videos about highly composite numbers (“Why 5040 is cool”) you’ll see that numbers that factor into small primes (but not all 2’s, like 2048, or 2’s and 3’s) tend to have the largest number of divisors and the highest sigma values.

Going one more year out, 2017 is prime. Going back a little, 2012 = 2 * 2 * 503, and 2011 was prime. (Thanks to for a lot of the data for all of these statements!)

Hope that was fun.

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