52

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.

Also see

 * Length of God's Algorithm for Sam Loyd's Fifteen Puzzle