Equidigital number

In number theory, an equidigital number is a natural number in a given number base that has the same number of digits as the number of digits in its prime factorization in the given number base, including exponents but excluding exponents equal to 1.^[1] For example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigital numbers (sequence A046758 in the OEIS). All prime numbers are equidigital numbers in any base.

Demonstration, with Cuisenaire rods, that the composite number 10 is equidigital: 10 has two digits, and 2 × 5 has two digits (1 is excluded)

A number that is either equidigital or frugal is said to be economical.

Mathematical definition

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

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

Properties

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

See also

• Extravagant number
• Frugal number
• Smith number