# 9

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## Number

$9$ (nine) is:

$3^2$

The $1$st odd prime power:
$9 = 3^2$

The $1$st power of $9$ after the zeroth $1$:
$9 = 9^1$

The larger of the $1$st pair of consecutive powerful numbers:
$8 = 2^3$, $9 = 3^2$

The $2$nd power of $3$ after $(1)$, $3$:
$9 = 3^2$

The sum of the first $2$ cubes:
$9 = 1^3 + 2^3$

The $2$nd square lucky number after $1$:
$1$, $9$, $\ldots$

The $2$nd Kaprekar number after $1$:
$9^2 = 81 \to 8 + 1 = 9$

The $2$nd integer after $1$ whose square has a divisor sum which is itself square:
$\map {\sigma_1} {9^2} = 11^2$

The $3$rd square number after $1$, $4$:
$9 = 3^2$
and therefore from Sum of Consecutive Triangular Numbers is Square, the sum of $2$ consecutive triangular numbers:
$9 = 3 + 6$

The $3$rd semiprime after $4$, $6$:
$9 = 3 \times 3$

The $3$rd Cullen number after $1$, $3$:
$9 = 2 \times 2^2 + 1$

The $3$rd square after $1$, $4$ which has no more than $2$ distinct digits

The sum of the first $3$ factorials:
$9 = 1! + 2! + 3!$

The $3$rd of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $\ldots$

The $4$th powerful number after $1$, $4$, $8$

The $4$th lucky number:
$1$, $3$, $7$, $9$, $\ldots$

The $4$th palindromic lucky number:
$1$, $3$, $7$, $9$, $\ldots$

The $4$th subfactorial after $0$, $1$, $2$:
$9 = 4! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \dfrac 1 {4!} }$

The $5$th trimorphic number after $1$, $4$, $5$, $6$:
$9^3 = 72 \mathbf 9$

The $5$th odd positive integer after $1$, $3$, $5$, $7$ such that all smaller odd integers greater than $1$ which are coprime to it are prime

The $5$th odd positive integer after $1$, $3$, $5$, $7$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

The $6$th integer after $0$, $1$, $3$, $5$, $7$ which is palindromic in both decimal and binary:
$9_{10} = 1001_2$

The $6$th positive integer after $2$, $3$, $4$, $7$, $8$ which cannot be expressed as the sum of distinct pentagonal numbers

The $7$th after $1$, $2$, $4$, $5$, $6$, $8$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

The $7$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$ which cannot be expressed as the sum of exactly $5$ non-zero squares

The $8$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

The $9$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$ such that $5^n$ contains no zero in its decimal representation:
$5^9 = 78 \, 125$

The $9$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$ such that both $2^n$ and $5^n$ have no zeroes:
$2^9 = 512$, $5^9 = 1 \, 953 \, 125$

The $9$th of the trivial $1$-digit pluperfect digital invariants after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$:
$9^1 = 9$

The $9$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$:
$9 = 1 \times 9$

The $9$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$:
$9 = 1 \times 9$

The $10$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ such that $2^n$ contains no zero in its decimal representation:
$2^9 = 512$

The $10$th integer after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ which is (trivially) the sum of the increasing powers of its digits taken in order:
$9^1 = 9$

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

Every positive integer can be expressed as the sum of at most $9$ positive cubes

The magic constant of a magic cube of order $2$ (if it were to exist), after $1$:
$9 = \ds \dfrac 1 {2^2} \sum_{k \mathop = 1}^{2^3} k = \dfrac {2 \paren {2^3 + 1} } 2$

In ternary:
$100_3 = 9_{10}$

### Arithmetic Functions on $9$

 $\ds \map \phi { 9 }$ $=$ $\ds 6$ $\phi$ of $9$

## Linguistic Note

Words derived from or associated with the number $9$ include:

nonagenarian: a person who has lived for $9$ decades