Definition:Multiplicative Persistence
Definition
Let $n \in \N$ be a natural number.
Let $n$ be expressed in decimal notation.
Multiply the digits of $n$ together.
Repeat with the answer, and again until a single digit remains.
The number of steps it takes to reach $1$ digit is called the multiplicative persistence of $n$.
Examples
10
$10$ is the smallest positive integer which has a multiplicative persistence of $1$.
25
$25$ is the smallest positive integer which has a multiplicative persistence of $2$.
39
$39$ is the smallest positive integer which has a multiplicative persistence of $3$.
77
$77$ is the smallest positive integer which has a multiplicative persistence of $4$.
679
$679$ is the smallest positive integer which has a multiplicative persistence of $5$.
$277 \, 777 \, 788 \, 888 \, 899$
$277 \, 777 \, 788 \, 888 \, 899$ is the smallest positive integer which has a multiplicative persistence of $11$.
Sequence of Smallest Arguments
Let $P \left({n}\right)$ denote the multiplicative persistence of a natural number $n$.
Let $a: \N \to \N$ be the partial mapping defined as:
- $\forall n \in \N: a \left({n}\right) = \text{the smallest $m \in \N$ such that $P \left({m}\right) = n$}$
The sequence of $a \left({n}\right)$ for successive $n$ begins as follows:
$n$ $a \left({n}\right)$ $0$ $0$ $1$ $10$ $2$ $25$ $3$ $39$ $4$ $77$ $5$ $679$ $6$ $6788$ $7$ $68 \, 889$ $8$ $2 \, 677 \, 889$ $9$ $26 \, 888 \, 999$ $10$ $3 \, 778 \, 888 \, 999$ $11$ $277 \, 777 \, 788 \, 888 \, 899$
This sequence is A003001 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
It is not known what $a \left({12}\right)$ is, but it is known to be greater than $10^{200}$.
Sources
- 1973: N.J.A. Sloane: The persistence of a number (J. Recreational Math. Vol. 6: pp. 97 – 98)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $10$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $277,777,788,888,899$