2016 (number)

2016 is the natural number following 2015 and preceding 2017.

Mathematics
2016 is the second-smallest Erdős–Nicolas number (after 24) because, while not perfect, 2016 is the sum of its first 31 divisors (up to and including 288). Furthermore, the sum of the following four divisors before its last (2016) is in equivalence with 2520, which is the first number to be divisible by all integers less than or equal to 10. It is one less than a prime number (2017), the 306th indexed prime.

2016 is a triangular number, where,

$$ 1 + 2 + 3 + \ldots + 63 = \binom{64}{2} = 2016.$$

It is also an hexagonal number, the fourteenth 24-gonal number, and in-turn the twenty-fourth generalized 28-gonal (icosioctagonal) number. 2016 has a total of 36 divisors, where 36 = 62 is the eighth triangular number (and 36 = 20 + 16).

2016 is the number of rooted Eulerian planar maps with five edges.

2016 is the smallest magic constant of a $$n \times n$$ magic square made of eight consecutive prime numbers.

2016 is the number of invertible $$2 \times 2$$ matrices $$\text{mod } 7.$$

2016 is coefficient $$44$$ of Eisenstein series $$E_2$$ (where 63 is the forty-fourth composite number), and Fourrier coefficient $$5$$ of $$E_{0,4}.$$

There are 2016 five-cubes in a nine-cube, and there are 2016 different lines determined by pair of vertices in a six-cube.

Friendly pair
2016 forms a friendly pair with 360, since they share the same abundancy: $$ \begin {align} \dfrac{\sigma(360)}{360} & = \dfrac{1170}{360} = \dfrac{13}{4},\text{ } \\ \dfrac{\sigma(2016)}{2016} & = \dfrac{6552}{2016} = \dfrac{13}{4} = 3.25. \\ \end {align} $$

The number 360 is itself a highly composite number, while 2016 — which is not strictly highly composite — is highly composite among the positive integers not divisible by 5 (cf. with highly composite numbers of class 4, where it is the eleventh element).

Amongst triangular numbers, 2016 is also highly composite, preceding the sequence $$\{1, 3, 6, 28, 36, 120, 300, 528, \mathbf{630}\}.$$

2016 is also the order of the 44th largest non-solvable group, where 360 is the 8th such order.

Other properties
$$2016 \times 2 + 1 = 4033 = 37 \times 109 $$ is a strong pseudoprime to base 2; aside from 2016, only five other numbers below 10,000 share this property (1023, 1638, 2340, 4160, and 7920).

2016 is the number of different products (including the empty product) of any subset of $$\{1, 2, 3, \ldots, 14\}.$$