21 (number)

	[[{{#expr: (floor({{{number}}} div {{{factor}}})) * {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}}}}]] [[{{#expr: (floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}{{{factor}}} div 10)}} (number)\|{{#switch:{{{1}}}\|-1={{#ifexpr:(floor({{{number}}} div 10)) = 0\|-1\|←}}\|10=→\|#default={{#expr:(floor({{{number}}} div {{{factor}}})) {{{factor}}}+({{{1}}}*{{{factor}}} div 10)}}}}]] List of numbers — Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twenty-one
Ordinal	21st; (twenty-first)
Factorization	3 × 7
Divisors	1, 3, 7, 21
Greek numeral	ΚΑ´
Roman numeral	XXI
Binary	101012
Ternary	2103
Quaternary	1114
Quinary	415
Senary	336
Octal	258
Duodecimal	1912
Hexadecimal	1516
Vigesimal	1120
Base 36	L36

21 (twenty-one) is the natural number following 20 and preceding 22.

In mathematics

21 is:

a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.^[1]
a Fibonacci number.^[2]
a Harshad number.^[3]
a Motzkin number.^[4]
a triangular number.^[5]
an octagonal number.^[6]
a composite number, its proper divisors being 1, 3 and 7.
the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21), making it a triangular number.
the sum of the sum of the divisors of the first 5 positive integers.
the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number.
a repdigit in base 4 (111₄).
the smallest natural number that is not close to a power of 2, 2ⁿ, where the range of closeness is ±n.
the smallest number of differently sized squares needed to square the square.^[7]
the largest n with this property: for any positive integers a,b such that a + b = n, at least one of ${\tfrac {a}{b}}$ and ${\tfrac {b}{a}}$ is a terminating decimal. See a brief proof below.

Note that a necessary condition for n is that for any a coprime to n, a and n - a must satisfy the condition above, therefore at least one of a and n - a must only have factor 2 and 5.

Let $A(n)$ donate the quantity of the numbers smaller than n that only have factor 2 and 5 and that are coprime to n, we instantly have ${\frac {\varphi (n)}{2}}<A(n)$ .

We can easily see that for sufficiently large n, $A(n)\sim {\frac {\log _{2}(n)\log _{5}(n)}{2}}={\frac {\ln ^{2}(n)}{2\ln(2)\ln(5)}}$ , but $\varphi (n)\sim {\frac {n}{e^{\gamma }\;\ln \ln n}}$ , $A(n)=o(\varphi (n))$ as n goes to infinity, thus ${\frac {\varphi (n)}{2}}<A(n)$ fails to hold for sufficiently large n.

In fact, For every n > 2, we have

A(n)<1+\log _{2}(n)+{\frac {3\log _{5}(n)}{2}}+{\frac {\log _{2}(n)\log _{5}(n)}{2}}

and

\varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}

so ${\frac {\varphi (n)}{2}}<A(n)$ fails to hold when n > 273 (actually, when n > 33).

Just check a few numbers to see that n = 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21.

21 appears in the Padovan sequence, preceded by the terms 9, 12, 16 (it is the sum of the first two of these).^[8]

In science

The atomic number of scandium.

Age 21

In several countries 21 is the age of majority. See also: Coming of age.
In all US states, 21 is the drinking age.
- However, in Puerto Rico and U.S. Virgin Islands, the drinking age is 18.
In California, Hawaii, New York, and New Jersey, 21 is the minimum age that one person may purchase cigarettes and other tobacco products.
In some countries it is the voting age.
In the United States, 21 is the age at which one can purchase multiple tickets to an R-rated film without providing Identifications. It is also the age to accompany one under the age of 17 as their parent or adult guardian for an R-rated movie.
In most US states, 21 is the minimum age at which a person may gamble or enter casinos.
In 2011, Adele named her second studio album 21, because of her age at the time.

In sports

Twenty-one is a variation of street basketball, in which each player, of which there can be any number, plays for himself only (i.e. not part of a team); the name comes from the requisite number of baskets.
In badminton, and table tennis (before 2001), 21 points are required to win a game.

In other fields

Building called "21" in Zlín, Czech Republic.

Detail of the building entrance

21 is:

The Twenty-first Amendment repealed the Eighteenth Amendment, thereby ending Prohibition.
The number of spots on a standard cubical (six-sided) die (1+2+3+4+5+6)
The number of firings in a 21-gun salute honoring Royalty or leaders of countries
21 Guns, a 2009 song by the punk-rock band Green Day
There are 21 trump cards of the tarot deck if one does not consider The Fool to be a proper trump card.
The standard TCP/IP port number for FTP connection
The Twenty-One Demands were a set of demands which were sent to the Chinese government by the Japanese government of Okuma Shigenobu in 1915
21 Demands of MKS led to the foundation of Solidarity in Poland.
In Israel, the number is associated with the profile 21 (the military profile designation granting an exemption from the military service)
21 grams is the weight of the soul, according to research by Duncan MacDougall, generally regarded as meaningless.
The number of the French department Côte-d'Or
The key value and highest-winning point total of the popular casino game Blackjack
The number of shillings in a guinea.
The number of solar rays in the flag of Kurdistan.

References

↑ "Sloane's A016105 : Blum integers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
↑ "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
↑ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
↑ "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
↑ "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
↑ "Sloane's A000567 : Octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
↑ C. J. Bouwkamp, and A. J. W. Duijvestijn, "Catalogue of Simple Perfect Squared Squares of Orders 21 Through 25." Eindhoven University of Technology, Nov. 1992.
↑ "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.

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[1] "Sloane's A016105 : Blum integers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.

[2] "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.

[3] "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.

[4] "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.

[5] "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.

[6] "Sloane's A000567 : Octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.

[7] C. J. Bouwkamp, and A. J. W. Duijvestijn, "Catalogue of Simple Perfect Squared Squares of Orders 21 Through 25." Eindhoven University of Technology, Nov. 1992.

[8] "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.