Multiplicative Digital Root

Consider the process of taking a number, multiplying its Digits, then multiplying the Digits of numbers derived from it, etc., until the remaining number has only one Digit. The number of multiplications required to obtain a single Digit from a number $n$ is called the Multiplicative Persistence of $n$, and the Digit obtained is called the multiplicative digital root of $n$.

For example, the sequence obtained from the starting number 9876 is (9876, 3024, 0), so 9876 has a Multiplicative Persistence of two and a multiplicative digital root of 0. The multiplicative digital roots of the first few positive integers 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 A031347 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.