Note: Posted to MO since it is unanswered in MSE.
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > f(a) + f(b)$ is impossible. Hence $f(ab)$ is either less than or equal to $f(a) + f(b)$. Loosely speaking if $f(ab) < f(a) + f(b)$, we are using more digits than the number itself hence we are wasting digits.
Definition: A number $n$ is said to be a wasteful number if for all its divisors $n = ab$, we have $f(ab) < f(a) + f(b)$.
E.g. The divisors of $4321$ are $[1, 29, 149, 4321]$ and clearly $4321$ is wasteful. But $4324$ is not a wasteful number since $4324 = 46 \times 94$ and $f(4324) = f(46) + f(94) = 4$. Trivially, all primes $p$ are wasteful since $f(p \times 1) < f(p) + f(1)$. The first few wasteful numbers are:
$$ 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99 $$
Update: Added graph for the density of wasteful numbers below $10^9$. There are $386000005$ wasteful numbers below $10^9$.
Question 1: Does the natural density of wasteful numbers exist? Experimental data for $n \le 10^9$ suggests that it oscillates in a regular pattern between $0.2$ and $0.4$.
Question 2: What are the necessary and sufficient conditions for a number to be wasteful?
Update: 7 Apr 2023 It was proved in MO that
Every wasteful number n must be its own lexicographically greatest divisor