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, no matter what base is used.
Let be a number base, and let be the number of digits in a natural number for base . A natural number has the integer factorisation
and is an extravagant number in base if
where is the p-adic valuation of .