8

Number
$8$ (eight) is:


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


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


 * The $2$nd octagonal number after $1$:
 * $8 = 1 + 7 = 2 \left({3 \times 2 - 2}\right)$


 * The $2$nd heptagonal pyramidal number after $1$:
 * $8 = 1 + 7 = \dfrac {2 \left({2 + 1}\right) \left({5 \times 2 - 2}\right)} 6$


 * The $6$th highly abundant number after $1$, $2$, $3$, $4$, $6$:
 * $\sigma \left({8}\right) = 15$


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


 * Equal to the sum of the digits of its cube:
 * $8^3 = 512$, while $5 + 1 + 2 = 8$


 * The $4$th almost perfect number after $1$, $2$, $4$:
 * $\sigma \left({8}\right) = 15 = 2 \times 8 - 1$


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


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


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


 * The base of the octal number system.


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


 * 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 $9$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 $3$rd element of the Fermat set after $1$, $3$


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


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


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


 * 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:
 * $8^1 = 8$


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


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


 * 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 $4$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 $n$:
 * $8^2 = \dbinom 7 0 + \dbinom 7 1 + \dbinom 7 2 + \dbinom 7 3$

Also see

 * Cube which is One Less than a Square
 * Cubic Fibonacci Numbers
 * Positive Integers Equal to Sum of Digits of Cube


 * Eight Convex Deltahedra


 * Definition:Octal Notation