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 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
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