Definition:Multiplicative Persistence

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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

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