63

Number

$63$ (sixty-three) is:

$3^2 \times 7$

The $3$rd of the $1$st ordered quadruple of consecutive integers that have divisor sums which are strictly increasing:

$\map {\sigma_1} {61} = 62$, $\map {\sigma_1} {62} = 96$, $\map {\sigma_1} {63} = 104$, $\map {\sigma_1} {64} = 127$

The $2$nd inconsummate number after $62$:

$\nexists n \in \Z_{>0}: n = 63 \times \map {s_{10} } n$

The $4$th Woodall number after $1$, $7$, $23$:

$63 = 4 \times 2^4 - 1$

The $15$th lucky number:

$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $\ldots$

One of the cycle of $5$ numbers to which Kaprekar's process on $2$-digit numbers converges:

$63 \to 27 \to 45 \to 09 \to 81 \to 63$