Extravagant_number Knowpia

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).^[1] For example, in base 10, 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers (sequence A046760 in the OEIS).

There are infinitely many extravagant numbers in every base.^[1]

Mathematical definition edit

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 edit

^ ^a ^b 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.

References edit

R.G.E. Pinch (1998), Economical Numbers.
Chris Caldwell, The Prime Glossary: extravagant number at The Prime Pages.

[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]

Summary

Mathematical definition edit

See also edit

Notes edit

References edit