# 27

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## Number

$27$ (twenty-seven) is:

$3^3$

The only cube number which is $2$ greater than a square:
$27 = 5^2 + 2$

The $1$st integer which has a reciprocal whose period is $3$:
$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$

The $1$st positive integer which is $3$ times the sum of its digits:
$27 = 3 \times \paren {2 + 7}$

The $1$st positive integer which is the sum of $3$ squares in $2$ ways:
$27 = 3^2 + 3^2 + 3^2 = 5^2 + 1^2 + 1^2$

The $2$nd of the only known pair of consecutive odd powerful numbers, the other being $25$:
$25 = 5^2$, $27 = 3^3$

The $2$nd element of the $1$st pair of integers $m$ whose values of $m \map \tau m$ is equal:
$18 \times \map \tau {18} = 108 = 27 \times \map \tau {27}$

The $2$nd odd cube number after $1$:
$27 = 3 \times 3 \times 3$

The $3$rd cube number after $1$, $8$:
$27 = 3 \times 3 \times 3$

The $3$rd power of $3$ after $(1)$, $3$, $9$:
$27 = 3^3$

The $3$rd Smith number after $4$, $22$:
$2 + 7 = 3 + 3 + 3 = 9$

The $5$th integer $m$ after $1$, $2$, $3$, $11$ such that $m! + 1$ is prime

The $6$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
$0$, $1$, $2$, $3$, $11$, $27$

The $7$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$

The $7$th and largest integer after $0$, $1$, $8$, $17$, $18$, $26$ equal to the sum of the digits of its cube:
$27^3 = 19 \, 683$, while $1 + 9 + 6 + 8 + 3 = 27$

The $11$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $\ldots$

The $14$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $\ldots$

The $16$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

The $16$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$, $24$:
$27 = 3 \times 9 = 3 \times \left({2 + 7}\right)$

The $19$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$, $18$, $19$, $24$, $25$ such that $2^n$ contains no zero in its decimal representation:
$2^{27} = 134 \, 217 \, 728$

One of the cycle of $5$ numbers to which Kaprekar's process on $2$-digit numbers converges:
$27 \to 45 \to 09 \to 81 \to 63 \to 27$

## Historical Note

The total score of all the coloured balls in snooker is $27$:

 $\displaystyle \text{Yellow}$ $:$ $\displaystyle 2$ $\displaystyle \text{Green}$ $:$ $\displaystyle 3$ $\displaystyle \text{Brown}$ $:$ $\displaystyle 4$ $\displaystyle \text{Blue}$ $:$ $\displaystyle 5$ $\displaystyle \text{Pink}$ $:$ $\displaystyle 6$ $\displaystyle \text{Black}$ $:$ $\displaystyle 7$

and $2 + 3 + 4 + 5 + 6 + 7 = 27$, which is one less than the $7$th triangular number $28$.