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 (sequence A046760 in the OEIS).

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

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

See also

• Equidigital number
• Frugal number

References

• R.G.E. Pinch (1998), Economical Numbers.
• Chris Caldwell, The Prime Glossary: extravagant number at The Prime Pages.