# 8

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## Number

$8$ (**eight**) is:

- $2^3$

- The base of the octal number system

- The number of pairs of twin primes less than $100$

- The $1$st power of $8$ after the zeroth $1$:
- $8 = 8^1$

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

- The $1$st number in English alphabetical sequence

- The $2$nd cube number after $1$:
- $8 = 2^3$

- The $2$nd of the only two cubic Fibonacci numbers after $1$:
- $8 = 3 + 5$

- The $2$nd octagonal number after $1$:
- $8 = 1 + 7 = 2 \paren {3 \times 2 - 2}$

- The $2$nd heptagonal pyramidal number after $1$:
- $8 = 1 + 7 = \dfrac {2 \paren {2 + 1} \paren {5 \times 2 - 2} } 6$

- The $2$nd Kaprekar triple after $1$:
- $8^3 = 512 \to 5 + 1 + 2 = 8$

- The $2$nd palindromic cube after $1$:
- $8 = 2^3$

- The number of different commutative binary operations that can be applied to a set with $2$ elements

- The $3$rd Dudeney number after $0$, $1$:
- $8^3 = 512$, while $5 + 1 + 2 = 8$

- The $3$rd power of $2$ after $(1)$, $2$, $4$:
- $8 = 2^3$

- The $3$rd powerful number after $1$, $4$

- The $3$rd element of the Fermat set after $1$, $3$

- The $3$rd of $6$ integers after $2$, $5$ which cannot be expressed as the sum of distinct triangular numbers

- The $4$th almost perfect number after $1$, $2$, $4$:
- $\map {\sigma_1} 8 = 15 = 2 \times 8 - 1$

- The $4$th of the $5$ known powers of $2$ whose digits are also all powers of $2$:
- $1$, $2$, $4$, $8$, $\ldots$

- The $4$th even number after $2$, $4$, $6$ which cannot be expressed as the sum of $2$ composite odd numbers

- The $4$th integer $m$ after $0$, $1$, $2$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
- $8^2 = \dbinom 7 0 + \dbinom 7 1 + \dbinom 7 2 + \dbinom 7 3$

- The $5$th positive integer which is not the sum of $1$ or more distinct squares:
- $2$, $3$, $6$, $7$, $8$, $\ldots$

- The $5$th integer after $0$, $1$, $2$, $4$ which is palindromic in both decimal and ternary:
- $8_{10} = 22_3$

- The $5$th positive integer after $2$, $3$, $4$, $7$ which cannot be expressed as the sum of distinct pentagonal numbers

- The $6$th Fibonacci number after $1$, $1$, $2$, $3$, $5$:
- $8 = 3 + 5$

- The $6$th highly abundant number after $1$, $2$, $3$, $4$, $6$:
- $\map {\sigma_1} 8 = 15$

- The $6$th Ulam number after $1$, $2$, $3$, $4$, $6$:
- $8 = 2 + 6$

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

- The $6$th positive integer after $1$, $2$, $3$, $4$, $6$ such that all smaller positive integers coprime to it are prime

- The $6$th integer $n$ after $3$, $4$, $5$, $6$, $7$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
- $8! - 7! + 6! - 5! + 4! - 3! + 2! - 1! = 35 \, 899$

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

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

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

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

- The $8$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

- The $9$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$ such that $2^n$ contains no zero in its decimal representation:
- $2^8 = 256$

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

### Arithmetic Functions on $8$

\(\ds \map {\sigma_0} { 8 }\) | \(=\) | \(\ds 4\) | $\sigma_0$ of $8$ | |||||||||||

\(\ds \map \phi { 8 }\) | \(=\) | \(\ds 4\) | $\phi$ of $8$ | |||||||||||

\(\ds \map {\sigma_1} { 8 }\) | \(=\) | \(\ds 15\) | $\sigma_1$ of $8$ |

## Also see

### Previous in Sequence: $1$

*Previous*: Cubic Fibonacci Numbers*Previous ... Next*: Dudeney Number*Previous ... Next*: Octagonal Number*Previous ... Next*: Heptagonal Pyramidal Number*Previous ... Next*: Cube Number*Previous ... Next*: Kaprekar Triple*Previous ... Next*: Sequence of Powers of 8*Previous ... Next*: Sequence of Palindromic Cubes*Previous ... Next*: Count of Commutative Binary Operations on Set

### Previous in Sequence: $2$

*Previous ... Next*: Square Formed from Sum of 4 Consecutive Binomial Coefficients*Previous ... Next*: Number of Twin Primes less than Powers of 10

### Previous in Sequence: $3$

*Previous ... Next*: Fermat Set

### Previous in Sequence: $4$

*Previous ... Next*: Powerful Number

*Previous ... Next*: Palindromes in Base 10 and Base 3*Previous ... Next*: Powers of 2 whose Digits are Powers of 2

### Previous in Sequence: $5$

*Previous ... Next*: Integers not Sum of Distinct Triangular Numbers*Previous ... Next*: Fibonacci Number

### Previous in Sequence: $6$

*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Ulam Number*Previous ... Next*: Integers such that all Coprime and Less are Prime

### Previous in Sequence: $7$

*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Zuckerman Number*Previous ... Next*: Harshad Number*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1

*Previous ... Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum

### Next in Sequence: $9$ and above

## Historical Note

In the mundane world, the most immediately relevant appearance of the number $8$ is the number of notes in an octave of a diatonic scale.

## Linguistic Note

Words derived from or associated with the number $8$ include:

**octopus**: a sea creature with $8$ feet

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $8$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $8$

Categories:

- Dudeney Numbers/Examples
- Octagonal Numbers/Examples
- Pyramidal Numbers/Examples
- Cube Numbers/Examples
- Kaprekar Numbers/Examples
- Powers of 8/Examples
- Count of Commutative Binary Operations on Set/Examples
- Powerful Numbers/Examples
- Almost Perfect Numbers/Examples
- Powers of 2/Examples
- Fibonacci Numbers/Examples
- Highly Abundant Numbers/Examples
- Ulam Numbers/Examples
- Pluperfect Digital Invariants/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Specific Numbers
- 8