multiplicative digital root

Given an integer $m$ consisting of $k$ digits $d_1,\mathrm{\dots },d_k$ in base $b$, let

$$j=\prod _{i=1}^k{d}_i,$$

then repeat this operation on the digits of $j$ until . This stores in $j$ the multiplicative digital root of $m$. The number of iterations of the multiplication operation is called the multiplicative persistence of $m$.