Natural number
Cardinal two thousand sixteen
Ordinal 2016th (two thousand sixteenth)
Factorization 25 × 32 × 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
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 ).
[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 306 th indexed prime.
[2]
2016 is a
triangular number ,
[3] where,
1
+
2
+
3
+
…
+
63
=
(
64
2
)
=
2016.
{\displaystyle 1+2+3+\ldots +63={\binom {64}{2}}=2016.}
It is also an hexagonal number,
[4] the fourteenth 24-
gonal number ,
[5] and in-turn the twenty-fourth generalized 28-gonal (icosioctagonal ) number.
[6] 2016 has a total of 36
divisors , where 36 =
6 2 is the eighth triangular number (and
36 =
20 +
16 ).
[3]
2016 is the number of rooted
Eulerian planar maps with five
edges .
[7]
2016 is the smallest
magic constant of a
n
×
n
{\displaystyle n\times n}
magic square made of eight consecutive prime numbers.
[8]
2016 is the number of
invertible
2
×
2
{\displaystyle 2\times 2}
matrices
mod
7.
{\displaystyle {\text{mod }}7.}
[9]
2016 is coefficient
44
{\displaystyle 44}
of
Eisenstein series
E
2
{\displaystyle E_{2}}
[10] (where 63 is the forty-fourth
composite number ),
[11] and
Fourrier coefficient
5
{\displaystyle 5}
of
E
0
,
4
.
{\displaystyle E_{0,4}.}
[12]
There are 2016
five-cubes in a
nine-cube , and there are 2016 different lines determined by pair of
vertices in a
six-cube .
[13]
Friendly pair
2016 forms a
friendly pair with
360 , since they share the same
abundancy :
σ
(
360
)
360
=
1170
360
=
13
4
,
σ
(
2016
)
2016
=
6552
2016
=
13
4
=
3.25.
{\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 ,
[14] 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
,
630
}
.
{\displaystyle \{1,3,6,28,36,120,300,528,\mathbf {630} \}.}
[15]
2016 is also the order of the 44th largest
non-solvable group , where 360 is the 8th such order.
[16]
Other properties
2
11
−
2
5
=
2048
−
32
=
2016
{\displaystyle 2^{11}-2^{5}=2048-32=2016}
(the difference between
powers of two ),
8
!
=
20
×
2016
=
40320
{\displaystyle 8!=20\times 2016=40320}
(or eight
factorial ),
2016
17
+
1
2017
{\displaystyle {{2016^{17}+1} \over 2017}}
is prime (since 2017 is similarly prime, 201617 + 1 is a
semiprime ).
[17]
2016
×
2
+
1
=
4033
=
37
×
109
{\displaystyle 2016\times 2+1=4033=37\times 109}
is a
strong pseudoprime to base 2;
[18] 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
,
…
,
14
}
.
{\displaystyle \{1,2,3,\ldots ,14\}.}
[19]
References
^
Sloane, N. J. A. (ed.).
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^
Sloane, N. J. A. (ed.).
"Sequence A000040 (The prime numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
a
b
Sloane, N. J. A. (ed.).
"Sequence A000217 (Triangular numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A000384 (Hexagonal numbers: a(n) = n*(2*n-1))" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A051876 (24-gonal numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A303812 (Generalized 28-gonal (or icosioctagonal) numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
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.
^
Sloane, N. J. A. (ed.).
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On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
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.
^
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.
^
Sloane, N. J. A. (ed.).
"Sequence A002808 (The composite numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A035016 (Fourier coefficients of E_{0,4)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. }
^
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.
^
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.
^
Sloane, N. J. A. (ed.).
"Sequence A076711 (Highly composite triangular numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
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.
^
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.
^
Sloane, N. J. A. (ed.).
"Sequence A001262 (Strong pseudoprimes to base 2)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
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. }