MathDB

Let n be a non-negative integer. Define the decimal digit product

D(n)

inductively as follows:
- If

n

has a single decimal digit, then let

D(n) = n

.
- Otherwise let

D(n) = D(m)

, where

m

is the product of the decimal digits of

n

.
Let

P_k(1)

be the probability that

D(i) = 1

where

i

is chosen uniformly randomly from the set of integers between 1 and

k

(inclusive) whose decimal digit products are not 0.
Compute

\displaystyle\lim_{k\to\infty} P_k(1)

.
proposed by the ICMC Problem Committee

Problem Statement