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 in every base.^[1]

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 prime factorisation

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

where ${\displaystyle v_{p}(n)}$ is the p-adic valuation of ${\displaystyle n}$, and ${\displaystyle n}$ is an extravagant number in base ${\displaystyle b}$ if

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

• Equidigital number
• Frugal number

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

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Descartes Refactorable Superperfect