Smallest Arguments for given Multiplicative Persistence

Sequence

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

• 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): $277,777,788,888,899$