← 26 27 28 →
Cardinaltwenty-seven
Ordinal27th
Factorization33
Divisors1, 3, 9, 27
Greek numeralΚΖ´
Roman numeralXXVII
Binary110112
Ternary10003
Senary436
Octal338
Duodecimal2312

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 ${\displaystyle ^{2}3=3^{3}=3\times 3\times 3}$, 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 ${\displaystyle (27,13,1,0)}$ of only one composite number, rooted in the 13-aliquot tree.[3]

The sum of the first four composite numbers is ${\displaystyle 4+6+8+9=27}$,[4] while the sum of the first four prime numbers is ${\displaystyle 2+3+5+7=17}$,[5] with 7 the fourth indexed prime.[6][a]

In the Collatz conjecture (i.e. the ${\displaystyle 3n+1}$ 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 ${\displaystyle {\tfrac {1}{7))}$ in a ${\displaystyle 6\times 6}$ square has a magic constant of 27.

Including the null-motif, there are 27 distinct hypergraph motifs.[14]

There are exactly twenty-seven straight lines on a smooth cubic surface,[15] which give a basis of the fundamental representation of Lie algebra ${\displaystyle \mathrm {E_{6)) }$.[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 ${\displaystyle \mathrm {F_{4)) .}$[19]

There are twenty-seven sporadic groups, if the non-strict group of Lie type ${\displaystyle \mathrm {T} }$ (with an irreducible representation that is twice that of ${\displaystyle \mathrm {F_{4)) }$ in 104 dimensions)[20] is included.[21]

In Robin's theorem for the Riemann hypothesis, twenty-seven integers fail to hold ${\displaystyle \sigma (n) for values ${\displaystyle n\leq 5040,}$ where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant; this hypothesis is true if and only if this inequality holds for every larger ${\displaystyle n.}$[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: ${\displaystyle 2+3+4+5+6+7=27}$.

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:

${\displaystyle 3.141\;592\;653\;589\;793\;238\;462\;643\;383\;{\color {red}27}9\ldots }$

If one starts counting with zero, 27 is the second self-locating string after 6, of only a few known.[27][28]

## In astrology

• 27 Nakṣatra or lunar mansions in Hindu astrology.

## 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.
• The current number of countries in the European Union, as of 2024.

## Notes

1. ^ 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]
2. ^ 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.
3. ^ 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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2. ^ a b Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
3. ^ a b Sloane, N. J. A., ed. (January 11, 1975). "Aliquot sequences". Mathematics of Computation. 29 (129). OEIS Foundation: 101–107. Retrieved October 31, 2023.
4. ^ Sloane, N. J. A. (ed.). "Sequence A151742 (Composite numbers which are the sum of four consecutive composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 2, 2023.
5. ^ Sloane, N. J. A. (ed.). "Sequence A007504 (Sum of the first n primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 2, 2023.
6. ^ a b Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
7. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
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11. ^
12. ^ Sloane, N. J. A. (ed.). "Sequence A077800 (List of twin primes {p, p+2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 8, 2023.
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