# 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): $277,777,788,888,899$