# 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 $2$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 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 $9$ |

## Also see

- 9 is Only Square which is Sum of 2 Consecutive Positive Cubes
- Nine Regular Polyhedra
- Dissection of Rectangle into 9 Distinct Integral Squares
- Nine Point Circle Theorem
- Divisibility by 9
- Hilbert-Waring Theorem for $3$rd Powers

### Previous in Sequence: $1$

*Previous ... Next*: Square Numbers whose Divisor Sum is Square*Previous ... Next*: Magic Constant of Magic Cube*Previous ... Next*: Kaprekar Number*Previous ... Next*: Sequence of Powers of 9

### Previous in Sequence: $2$

*Previous ... Next*: Subfactorial

### Previous in Sequence: $3$

*Previous ... Next*: Cullen Number*Previous ... Next*: Sequence of Powers of 3*Previous ... Next*: Sum of Sequence of Factorials

### Previous in Sequence: $4$

### Previous in Sequence: $6$

*Previous ... Next*: Semiprime Number*Previous ... Next*: Trimorphic Number

### Previous in Sequence: $7$

*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares*Previous ... Next*: Lucky Number*Previous ... Next*: Odd Integers whose Smaller Odd Coprimes are Prime*Previous ... Next*: Powers of 2 and 5 without Zeroes*Previous ... Next*: Palindromes in Base 10 and Base 2*Previous ... Next*: Sequence of Palindromic Lucky Numbers

### Previous in Sequence: $8$

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Harshad Number*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1

*Previous ... Next*: Zuckerman Number*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Powerful Number*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Consecutive Powerful Numbers

### Previous in Sequence: $45$

### Next in Sequence: $20$ and above

## Linguistic Note

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

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

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $9$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $9$

Categories:

- Square Numbers whose Divisor Sum is Square/Examples
- Kaprekar Numbers/Examples
- Powers of 9/Examples
- Subfactorials/Examples
- Cullen Numbers/Examples
- Powers of 3/Examples
- Square Numbers/Examples
- Semiprimes/Examples
- Trimorphic Numbers/Examples
- Lucky Numbers/Examples
- Harshad Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Zuckerman Numbers/Examples
- Powerful Numbers/Examples
- Pluperfect Digital Invariants/Examples
- Specific Numbers
- 9