# 8

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## Number

$8$ (eight) is:

$2^3$

The base of the octal number system

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 number of pairs of twin primes less than $10^2$

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 $8$

## 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