Frugal number

In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents).[1] For example, in base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers (sequence A046759 in the OEIS), and in base 2, thirty-two is a frugal number, since 100000 = 10101.

The term economical number has been used about a frugal number, but also about a number which is either frugal or equidigital.

Mathematical definition

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 integer factorisation

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

and is an frugal number in base ${\displaystyle b}$ if

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

where ${\displaystyle v_{p}(n)}$ is the p-adic valuation of ${\displaystyle n}$.