52

Number
$52$ (fifty-two) is:


 * $2^2 \times 13$


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


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


 * The $4$th term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the $\sigma$ function $\sigma \left({n}\right)$ for any $n$:
 * $\left({49, 50, 51, 52, 53}\right)$


 * 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