Extravagant number

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

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).{{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, 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers {{OEIS|id=A046760}}.

There are infinitely many extravagant numbers in every base.

Mathematical definition

Let $b > 1$ be a 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 extravagant number in base $b$ if

: $K_b(n) < \sum_{{\stackrel{p \,\mid\, n}{p\text{ prime}}}} K_b(p) + \sum_{{\stackrel{p^2 \,\mid\, n}{p\text{ prime}}}} K_b(v_p(n)).$

Notes

References

R.G.E. Pinch (1998), [https://arxiv.org/abs/math/9802046 Economical Numbers].
Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=ExtravagantNumber The Prime Glossary: extravagant number] at The Prime Pages.

Category:Base-dependent integer sequences

Extravagant number

Mathematical definition

See also

Notes

References