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equidigital number

{{Short description|Number that has the same number of digits as the number of digits in its prime factorization}}

File:Composite number Cuisenaire rods 10.svg, that the composite number 10 is equidigital: 10 has two digits, and 2 × 5 has two digits (1 is excluded)]]

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.{{cite book |title=The universal book of mathematics: from Abracadabra to Zeno's paradoxes |last=Darling |first=David J. |year=2004 |publisher=John Wiley & Sons |isbn=978-0-471-27047-8 |page=102 |url=https://books.google.com/books?id=nnpChqstvg0C&pg=PA102 }} For example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigital numbers {{OEIS|id=A046758}}. All prime numbers are equidigital numbers in any base.

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

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.

See also

Notes

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References

  • R.G.E. Pinch (1998), [https://arxiv.org/abs/math/9802046 Economical Numbers].

{{Divisor classes}}

{{Classes of natural numbers}}

Category:Integer sequences

Category:Base-dependent integer sequences