Frugal number

In

, thirty-two is a frugal number, since 32 = 2⁵ is written in base 2 as 100000 = 10¹⁰¹.

The term economical number has been used for a frugal number, but also for 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 prime factorisation

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

where ${\displaystyle v_{p}(n)}$ is the p-adic valuation of ${\displaystyle n}$, and ${\displaystyle n}$ is an frugal number in base ${\displaystyle b}$ if

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

See also

• Equidigital number
• Extravagant number

Notes

References

• R.G.E. Pinch (1998), Economical Numbers