Multiplicative Digital Root
Consider the process of taking a number, multiplying its Digits, then multiplying the Digitsof numbers derived from it, etc., until the remaining number has only one Digit. The number of multiplications requiredto obtain a single Digit from a number 
is called the Multiplicative Persistence of , and the Digitobtained is called the multiplicative digital root of .
For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has a MultiplicativePersistence of two and a multiplicative digital root of 0. The multiplicative digital roots of the first few positiveintegers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2,... (Sloane's A031347).
See also Additive Persistence, Digitaddition, Digital Root, Multiplicative Persistence
References
Sloane, N. J. A. Sequence A031347in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.