Extravagant number

In number theory, an extravagant number (also known as a wasteful number) is a natural number in a given number base that has fewer digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 4 = 22, 6 = 2×3, 8 = 23, and 9 = 32 are extravagant numbers.

There are infinitely many extravagant numbers in every base.

Mathematical definition
Let $$b > 1$$ be a number base, and let $$K_b(n) = \lfloor \log_{b}{n} \rfloor + 1$$ be the number of digits in a natural number $$n$$ for base $$b$$. A natural number $$n$$ has the prime factorisation
 * $$n = \prod_{\stackrel{p \,\mid\, n}{p\text{ prime}}} p^{v_p(n)}$$

where $$v_p(n)$$ is the p-adic valuation of $$n$$, and $$n$$ is an extravagant number in base $$b$$ if
 * $$K_b(n) < \sum_ K_b(p) + \sum_ K_b(v_p(n)).$$