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).^[1] For example, in base 10, 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers (sequence A046760 in the OEIS).

There are infinitely many extravagant numbers, no matter what base is used.^[1]

Mathematical definition

Let ${\displaystyle b>1}$ be a number base, and let ${\displaystyle K_{b}(n)=\lfloor \log _{b}{n}\rfloor +1}$ be the number of digits in a natural number ${\displaystyle n}$ for base ${\displaystyle b}$. A natural number ${\displaystyle n}$ has the integer factorisation

${\displaystyle n=\prod _{\stackrel {p\mid n}{p{\text{ prime))))p^{v_{p}(n)))$

and is an extravagant number in base ${\displaystyle b}$ if

${\displaystyle K_{b}(n)<\sum _{\stackrel {p\mid n}{p{\text{ prime))))K_{b}(p)+\sum _{\stackrel {p^{2}\mid n}{p{\text{ prime))))K_{b}(v_{p}(n))}$

where ${\displaystyle v_{p}(n)}$ is the p-adic valuation of ${\displaystyle n}$.