9
Jump to navigation
Jump to search
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 $2$nd positive integer after $6$ whose cube can be expressed as the sum of $3$ positive cube numbers:
- $9^3 = 1^3 + 6^3 + 8^3$
- 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 and does not end in $0$:
- $9 = 3^2$
- 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: Cubes which are Sum of Three Cubes
- 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