Extravagant number


In number theory, an extravagant 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. For example, in base 10, 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers.
There are infinitely many extravagant numbers, no matter what base is used.

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 integer factorisation
and is an extravagant number in base if
where is the p-adic valuation of.