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 be a number base, and let be the number of digits in a natural number for base . A natural number has the prime factorisation
where is the p-adic valuation of , and is an extravagant number in base if
See also Equidigital number Frugal number
Notes
References R.G.E. Pinch (1998), Economical Numbers. Chris Caldwell, The Prime Glossary: extravagant number at The Prime Pages.