Equidigital number

Last updated December 13, 2024

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.

Mathematical definition

Let $b>1$ be the number base, and let $K_{b}(n)=\lfloor \log _{b}{n}\rfloor +1$ be the number of digits in a natural number $n$ for base $b$ . A natural number $n$ has the prime factorisation

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

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

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.

Notes

↑ Darling, David J. (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley & Sons. p. 102. ISBN 978-0-471-27047-8.

Related Research Articles

In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.

The number $e$ is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted $. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.$

In mathematics, the factorial of a non-negative integer $,$ denoted by $,$ is the product of all positive integers less than or equal to $.$ The factorial of also equals the product of $with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product.$

In mathematics, the logarithm to base $b$ is the inverse function of exponentiation with base $b$ . That means that the logarithm of a number $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$ . For example, since $1000 = 10 3$ , the logarithm base $of 1000 is 3, or log 10 (1000) = 3 . The logarithm of x to base b is denoted as log b (x), or without parentheses, log b x . When the base is clear from the context or is irrelevant it is sometimes written log x .$

The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted $.$

<span class="mw-page-title-main">Euler's totient function</span> Number of integers coprime to and less than n

In number theory, Euler's totient function counts the positive integers up to a given integer $n$ that are relatively prime to $n$ . It is written using the Greek letter phi as $or, and may also be called Euler's phi function . In other words, it is the number of integers k in the range 1 \leq k \leq n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n .$

The tables contain the prime factorization of the natural numbers from 1 to 1000.

A power of two is a number of the form $2 n$ where $n$ is an integer, that is, the result of exponentiation with number two as the base and integer $n$ as the exponent.

In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed.

In mathematics, a harshad number in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base $n$ are also known as $n$ -harshad numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.

In number theory, a self number or Devlali number in a given number base $is a natural number that cannot be written as the sum of any other natural number and the individual digits of . 20 is a self number, because no such combination can be found. 21 is not, because it can be written as 15 + 1 + 5 using n = 15. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar.$

In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number $would be$

In number theory, a Dudeney number in a given number base $is a natural number equal to the perfect cube of another natural number such that the digit sum of the first natural number is equal to the second. The name derives from Henry Dudeney, who noted the existence of these numbers in one of his puzzles, Root Extraction, where a professor in retirement at Colney Hatch postulates this as a general method for root extraction.$

In number theory, a narcissistic number in a given number base $is a number that is the sum of its own digits each raised to the power of the number of digits.$

A sum-product number in a given number base $is a natural number that is equal to the product of the sum of its digits and the product of its digits.$

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). For example, in base 10, 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers (sequence A046760 in the OEIS).

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). For example, in base 10, 125 = 5³, 128 = 2⁷, 243 = 3⁵, and 256 = 2⁸ are frugal numbers (sequence A046759 in the OEIS). The first frugal number which is not a prime power is 1029 = 3 × 7³. In base 2, thirty-two is a frugal number, since 32 = 2⁵ is written in base 2 as 100000 = 10¹⁰¹.

In number theory and mathematical logic, a Meertens number in a given number base $is a natural number that is its own Gödel number. It was named after Lambert Meertens by Richard S. Bird as a present during the celebration of his 25 years at the CWI, Amsterdam.$

In number theory, the prime omega functions $and count the number of prime factors of a natural number Thereby counts each distinct prime factor, whereas the related function counts the total number of prime factors of honoring their multiplicity. That is, if we have a prime factorization of of the form for distinct primes, then the respective prime omega functions are given by and . These prime factor counting functions have many important number theoretic relations.$

The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number $, or equivalently the Dirichlet convolution of an arithmetic function with one:$

References

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

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Darling102-1] Darling, David J. (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley & Sons. p. 102. ISBN 978-0-471-27047-8.

[1]

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect