# 52

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## Number

$52$ (fifty-two) is:

$2^2 \times 13$

The $4$th term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
$\tuple {49, 50, 51, 52, 53}$

The $3$rd untouchable number after $2$, $5$

The $5$th Bell number after $(1)$, $1$, $2$, $5$, $15$

The $5$th noncototient after $10$, $26$, $34$, $50$:
$\nexists m \in \Z_{>0}: m - \map \phi m = 52$
where $\map \phi m$ denotes the Euler $\phi$ function

The length of God's Algorithm for Sam Loyd's Fifteen Puzzle.

### Arithmetic Functions on $52$

 $\ds \map {\sigma_0} { 52 }$ $=$ $\ds 6$ $\sigma_0$ of $52$ $\ds \map \phi { 52 }$ $=$ $\ds 24$ $\phi$ of $52$ $\ds \map {\sigma_1} { 52 }$ $=$ $\ds 98$ $\sigma_1$ of $52$

## Historical Note

There are two particular cultural significances for the number $52$:

There are $52$ weeks (plus $1$ or $2$ days) in a year, divided into $4$ seasons of $13$ weeks each.
There are $52$ cards in a standard deck divided into $4$ suits of $13$ cards each.