Numbers for which Euler Phi Function equals Product of Digits

Theorem

The sequence of positive integers $n$ for which $\map \phi n$ is equal to the product of the digits of $n$ begins:

$1, 24, 26, 87, 168, 388, 594, 666, 1998, 2688, 5698, 5978, 6786, 7888, 68 \, 796$

This sequence is A058627 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

It is known that this sequence is finite, but it is unknown whether a $16^{\text {th}}$ term exists.

Proof

\(\ds \map \phi 1\)

\(=\)

\(\ds 1\)

$\phi$ of $1$

\(\ds \map \phi {24}\)

\(=\)

\(\ds 8\)

$\phi$ of $24$

\(\ds \)

\(=\)

\(\ds 2 \times 4\)

\(\ds \map \phi {26}\)

\(=\)

\(\ds 12\)

$\phi$ of $26$

\(\ds \)

\(=\)

\(\ds 2 \times 6\)

\(\ds \map \phi {87}\)

\(=\)

\(\ds 56\)

$\phi$ of $87$

\(\ds \)

\(=\)

\(\ds 8 \times 7\)

\(\ds \map \phi {168}\)

\(=\)

\(\ds 48\)

$\phi$ of $168$

\(\ds \)

\(=\)

\(\ds 1 \times 6 \times 8\)

and so on.

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