Equidigital number

Let ${\displaystyle b>1}$ be the 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 equidigital number in base ${\displaystyle b}$ if

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

• Every prime number is equidigital. This also proves that there are infinitely many equidigital numbers.

• Extravagant number
• Frugal number
• Smith number

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