2016 (number)

2016 is the natural number following 2015 and preceding 2017.

← 2015 2016 2017 →
List of numbers Integers ← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinal	two thousand sixteen
Ordinal	2016th (two thousand sixteenth)
Factorization	2^5 × 3^2 × 7
Divisors	1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016
Greek numeral	,ΒΙϚ´
Roman numeral	MMXVI
Binary	111111000002
Ternary	22022003
Senary	132006
Octal	37408
Duodecimal	120012
Hexadecimal	7E016

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).^[1] 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.^[2]

2016 is a triangular number,^[3] where,

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

 

It is also the fourteenth 24-gonal number,^[4] and in-turn the twenty-fourth generalized 28-gonal (icosioctagonal) number.^[5] 2016 has a total of 36 divisors, where 36 = 6² is the eighth triangular number (and 36 = 20 + 16).^[3]

2016 is the number of rooted Eulerian planar maps with five edges.^[6]

2016 is the smallest magic constant of a ${\displaystyle n\times n}$  magic square made of eight consecutive prime numbers.^[7]

2016 is the number of invertible ${\displaystyle 2\times 2}$  matrices ${\displaystyle {\text{mod }}7.}$ ^[8]

2016 is coefficient ${\displaystyle 44}$  of Eisenstein series ${\displaystyle E_{2}}$ ^[9] (where 63 is the forty-fourth composite number),^[10] and Fourrier coefficient ${\displaystyle 5}$  of ${\displaystyle E_{0,4}.}$ ^[11]

There are 2016 five-cubes in a nine-cube, and there are 2016 different lines determined by pair of vertices in an six-cube.^[12]

Friendly pair

2016 forms a friendly pair with 360, since they share the same abundancy:

$${\displaystyle {\begin{aligned}{\dfrac {\sigma (360)}{360}}&={\dfrac {1170}{360}}={\dfrac {13}{4}},{\text{ }}\\{\dfrac {\sigma (2016)}{2016}}&={\dfrac {6552}{2016}}={\dfrac {13}{4}}=3.25.\\\end{aligned}}}$$

 

The number 360 is itself a highly composite number,^[13] 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 ${\displaystyle \{1,3,6,28,36,120,300,528,{\mathbf {630}}\}.}$ ^[14]

2016 is also the order of the 44th largest non-solvable group, where 360 is the 8th such order.^[15]

Other properties

• ${\displaystyle 2^{11}-2^{5}=2048-32=2016}$  (the difference between powers of two),
• ${\displaystyle 8!=20\times 2016=40320}$  (or eight factorial),
• ${\displaystyle {{2016^{17}+1} \over 2017}}$  is prime (since 2017 is similarly prime, 2016¹⁷ + 1 is a semiprime).^[16]

${\displaystyle 2016\times 2+1=4033=37\times 109}$  is a strong pseudoprime to base 2;^[17] 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 ${\displaystyle \{1,2,3,\ldots ,14\}.}$ ^[18]

References

1. Sloane, N. J. A. (ed.). "Sequence A194472 (Erdős-Nicolas numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
2. Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
3. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
4. Sloane, N. J. A. (ed.). "Sequence A051876 (24-gonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
5. Sloane, N. J. A. (ed.). "Sequence A303812 (Generalized 28-gonal (or icosioctagonal) numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
6. Sloane, N. J. A. (ed.). "Sequence A069720 (a(n) equal to 2^(n-1)*binomial(2n-1, n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
7. Sloane, N. J. A. (ed.). "Sequence A073520 (Smallest magic constant for any n X n magic square made from consecutive primes, or 0 if no such magic square exists.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
8. Sloane, N. J. A. (ed.). "Sequence A000252 (Number of invertible 2 X 2 matrices mod n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
9. Sloane, N. J. A. (ed.). "Sequence A006352 (Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
10. Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
11. Sloane, N. J. A. (ed.). "Sequence A035016 (Fourier coefficients of E_{0,4}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
12. Sloane, N. J. A. (ed.). "Sequence A006516 (a(n) equal to 2^(n-1)*(2^n - 1), n greater than or equal to 0.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
13. Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
14. Sloane, N. J. A. (ed.). "Sequence A076711 (Highly composite triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
15. Sloane, N. J. A. (ed.). "Sequence A056866 (Orders of non-solvable groups, i.e., numbers that are not solvable numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
16. Sloane, N. J. A. (ed.). "Sequence A104494 (Positive integers n such that n^17 + 1 is semiprime (A001358).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
17. Sloane, N. J. A. (ed.). "Sequence A001262 (Strong pseudoprimes to base 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.
18. Sloane, N. J. A. (ed.). "Sequence A060957 (Number of different products (including the empty product) of any subset of {1, 2, 3, ..., n}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27.