27

== 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 $m \, \map {\sigma_0} m$ is equal:
::$18 \times \map {\sigma_0} {18} = 108 = 27 \times \map {\sigma_0} {27}$
: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 smallest positive integer the decimal expansion of whose reciprocal has a period of $3$:
::$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$
:The $4$th after $0$, $2$, $3$ of the $6$ integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square
::$25^3 + 26^3 + 27^3 + 28^3 + 29^3 = 99 \, 225 = 315^2$
: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 Dudeney number after $0$, $1$, $8$, $17$, $18$, $26$:
::$27^3 = 19 \, 683$, 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 $14$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $\ldots$
: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 $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$, $24$:
::$27 = 3 \times 9 = 3 \times \paren {2 + 7}$
: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:
::$2^{27} = 134 \, 217 \, 728$
:The $21$st (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $20$, $21$, $24$, $25$, $26$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:One of the cycle of $5$ numbers to which Kaprekar's process on $2$-digit numbers converges:
::$27 \to 45 \to 09 \to 81 \to 63 \to 27$
== Also see ==
* Consecutive Odd Powerful Numbers
* Square which is 2 Less than Cube
* 27 is Smallest Number whose Period of Reciprocal is 3
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== Historical Note ==
== Sources ==
* : $27$
* : $27$
Category:Specific Numbers
Category:27
Category:Powerful Numbers