Natural number
27 (twenty-seven ; Roman numeral XXVII ) is the natural number following
26 and preceding
28 .
Mathematics
Twenty-seven is the
cube of
3 , or three
tetrated
2
3
=
3
3
=
3
×
3
×
3
{\displaystyle ^{2}3=3^{3}=3\times 3\times 3}
, divisible by the number of
prime numbers below it (
nine ).
The first non-trivial
decagonal number is 27.
[1]
27 has an
aliquot sum of
13
[2] (the sixth prime number) in the
aliquot sequence
(
27
,
13
,
1
,
0
)
{\displaystyle (27,13,1,0)}
of only one composite number, rooted in the
13 -aliquot tree.
[3]
The sum of the first four composite numbers is
4
+
6
+
8
+
9
=
27
{\displaystyle 4+6+8+9=27}
,
[4] while the sum of the first four prime numbers is
2
+
3
+
5
+
7
=
17
{\displaystyle 2+3+5+7=17}
,
[5] with 7 the fourth indexed prime.
[6]
[a]
In the
Collatz conjecture (i.e. the
3
n
+
1
{\displaystyle 3n+1}
problem), a starting value of 27 requires
3 ×
37 =
111 steps to reach 1, more than any smaller number.
[10]
[b]
27 is also the fourth
perfect totient number — as are all
powers of
3 — with its adjacent members
15 and
39 adding to twice 27.
[13]
[c]
A
prime reciprocal magic square based on multiples of
1
7
{\displaystyle {\tfrac {1}{7}}}
in a
6
×
6
{\displaystyle 6\times 6}
square has a
magic constant of 27.
Including the null-motif, there are 27 distinct
hypergraph motifs .
[14]
The
Clebsch surface , with 27 straight lines
There are exactly
twenty-seven straight lines on a smooth
cubic surface ,
[15] which give a basis of the
fundamental representation of
Lie algebra
E
6
{\displaystyle \mathrm {E_{6}} }
.
[16]
[17]
The unique simple formally real
Jordan algebra , the exceptional Jordan algebra of self-adjoint
3 by 3 matrices of
quaternions , is 27-dimensional;
[18] its
automorphism group is the 52-dimensional
exceptional Lie algebra
F
4
.
{\displaystyle \mathrm {F_{4}} .}
[19]
There are twenty-seven
sporadic groups , if the
non-strict group of Lie type
T
{\displaystyle \mathrm {T} }
(with an
irreducible representation that is twice that of
F
4
{\displaystyle \mathrm {F_{4}} }
in 104 dimensions)
[20] is included.
[21]
In
Robin's theorem for the
Riemann hypothesis , twenty-seven integers fail to hold
σ
(
n
)
<
e
γ
n
log
log
n
{\displaystyle \sigma (n)<e^{\gamma }n\log \log n}
for values
n
≤
5040
,
{\displaystyle n\leq 5040,}
where
γ
{\displaystyle \gamma }
is the
Euler–Mascheroni constant ; this hypothesis is true
if and only if this inequality holds for every larger
n
.
{\displaystyle n.}
[22]
[23]
[24]
Base-specific
In
decimal , 27 is the first
composite number not divisible by any of its digits, as well as:
the third
Smith number
[25] and sixteenth
Harshad number ,
[26]
the only positive integer that is three times the sum of its digits,
equal to the sum of the numbers between and including its digits:
2
+
3
+
4
+
5
+
6
+
7
=
27
{\displaystyle 2+3+4+5+6+7=27}
.
Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For example, 378, 783, and 837 are all divisible by 27.
In similar fashion, any multiple of 27 can be mirrored and spaced with a zero each for another multiple of 27 (i.e. 27 and 702, 54 and 405, and 378 and 80703 are all multiples of 27).
Any multiple of 27 with "000" or "999" inserted yields another multiple of 27 (20007, 29997, 50004, and 59994 are all multiples of 27).
In
senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if the last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513.
In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in
π :
3.141
592
653
589
793
238
462
643
383
27
9
…
{\displaystyle 3.141\;592\;653\;589\;793\;238\;462\;643\;383\;{\color {red}27}9\ldots }
If one starts counting with zero, 27 is the second self-locating string after
6 , of only a few known.
[27]
[28]
In science
Astronomy
The
Messier object
M27 , a
magnitude 7.5
planetary nebula in the
constellation
Vulpecula , also known as the
Dumbbell Nebula .
The
New General Catalogue object
NGC 27 , a
spiral galaxy in the
Andromeda constellation.
The
Saros number of the
solar eclipse series, which began on March 9, 1993, BCE and ended on April 16, 713 BCE.
[31] The duration of Saros series 27 was 1,280.1 years, and it contained 72 solar eclipses. Further, the Saros number of the
lunar eclipse series, which began on July 28, 1926, BCE and ended on January 23, 411 BCE.
[32] The duration of Saros series 27 was 1532.5 years, and it contained 86 lunar eclipses.
Electronics
In language and literature
In astrology
27
Nakṣatra or lunar mansions in Hindu astrology.
In art
Movies
Music
"27", a song by Fall Out Boy on the album
Folie à Deux
"27", a song by Passenger on the album
Whispers
"27", a song by Title Fight on the album Shed .
[34]
"
27 ", a song by Biffy Clyro on the album Blackened Sky
"27", a song on Machine Gun Kelly's album
Bloom
"27 Jennifers", a song by Mike Doughty on the album
Rockity Roll
27 , an album by South Korean singer Kim Sung-kyu
27 , an album by Argentine rock band Ciro y los Persas
Twenty-Seven , an album by
The Adicts .
French rapper
Kaaris ' signature number is 27, from his zip code
93270 .
"Twenty Seven Strangers" by Villagers.
27 , a Boston-based band
27 , an opera by composer Ricky Ian Gordon and librettist Royce Vavrek
27 Club , a list of popular musicians, artists, or actors who died at age 27
"Weird Al" Yankovic has a recurring joke involving the number 27, which is used in several songs.
Other
The Minneapolis-based artist
Deuce 7 (a.k.a. Deuce Seven, Twenty Seven, 27).
In sports
The value of all the colors in
snooker add up to 27.
The number of outs in a regulation baseball game for each team at all adult levels, including professional play, is 27.
The
New York Yankees have won 27
World Series championships, the most of any team in the
MLB .
In other fields
Twenty-seven is also:
A-27 , American attack aircraft.
The code for international direct-dial phone calls to
South Africa .
The name of a cigarette,
Marlboro Blend No. 27.
The number of the French department
Eure .
See also
Notes
^ Whereas the
composite index of 27 is
17
[7] (the
cousin prime to 13),
[8]
7 is the
prime index of 17.
[6] The sum 27 + 17 + 7 =
53 represents the sixteenth indexed prime (where 42 =
16 ). While 7 is the fourth prime number, the fourth composite number is
9 = 32 , that is also the
composite index of 16.
[9]
^ On the other hand, The reduced Collatz sequence of 27, that counts the number of prime numbers in its trajectory, is
41 .
[11] This count represents the thirteenth prime number, that is also in equivalence with the sum of members in the aliquot tree (27, 13, 1, 0).
[3]
[2] The next two larger numbers in the Collatz conjecture to require more than 111 steps to return to 1 are
54 and
55 Specifically, the fourteenth prime number
43 requires twenty-seven steps to reach 1. The sixth pair of
twin primes is (41, 43),
[12] whose respective prime
indices generate a sum of 27.
^ Also, 36 = 62 is the sum between
PTNs 39 – 15 = 24 and 3 + 9 = 12. In this sequence,
111 is the seventh PTN.
References
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Further reading
Wells, D.
The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987), p. 106.
External links
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000