# 625

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## Number

$625$ (six hundred and twenty-five) is:

$5^4$

The $1$st fourth power which is the sum of $5$ fourth powers:
$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$

The $1$st positive integer whose $4$th root equals the number of its divisors:
$\map \tau {625} = 5 = \sqrt [4] {625}$

The $4$th power of $5$ after $(1)$, $5$, $25$, $125$:
$625 = 5^4$

The $5$th fourth power after $1$, $16$, $81$, $256$:
$625 = 5 \times 5 \times 5 \times 5$

The $7$th automorphic number after $1$, $5$, $6$, $25$, $76$, $376$:
$625^2 = 390 \, \mathbf {625}$

The $21$st trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$, $51$, $75$, $76$, $99$, $125$, $249$, $251$, $375$, $376$, $499$, $501$, $624$:
$625^3 = 244 \, 140 \, \mathbf {625}$

The $21$st positive integer which cannot be expressed as the sum of a square and a prime:
$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $526$, $529$, $625$, $\ldots$

The $25$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $361$, $400$, $441$, $484$, $529$, $576$:
$625 = 25 \times 25$

The $42$nd powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $400$, $432$, $441$, $484$, $500$, $512$, $529$, $576$:
$625 = 5^4$

### Arithmetic Functions on $625$

 $\displaystyle \map \tau { 625 }$ $=$ $\displaystyle 5$ $\tau$ of $625$