# 512

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## Number

$512$ (five hundred and twelve) is:

$2^9$

In binary:
$1 \, 000 \, 000 \, 000$

In octal:
$1 \, 000$

The $3$rd cube after $0$, $1$ whose digits add up to its root:
$512 = 8^3$, while $5 + 1 + 2 = 8$

The $3$rd power of $8$ after $(1)$, $8$, $64$:
$512 = 8^3$

The $8$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$:
$512 = 8 \times 8 \times 8$

The $9$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$:
$512 = 2^9$

The $10$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$:
$\map {\sigma_1} {512} = 1023 = 2 \times 512 - 1$

The $10$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$ with $7$ or more prime factors:
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 2 \times 2}$

The index (after $2$, $3$, $6$, $30$, $75$, $81$, $115$, $123$, $249$, $362$, $384$, $462$) of the $13$th Woodall prime:
$512 \times 2^{512} - 1$

The $14$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$, $324$, $325$, $326$, $328$, $510$ that cannot be made into a prime number by changing just $1$ digit

The $25$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $384$, $400$, $416$, $432$, $448$, $480$, $504$ with $6$ or more prime factors:
$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 2 \times 2 \times 2}$

The $39$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $289$, $324$, $343$, $361$, $392$, $400$, $432$, $441$, $484$, $500$:
$512 = 2^9$