Extravagant number explained

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 .

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 factorisationn = \prod _{\stackrel{p} {p prime }} p^where v _{p(n)} is the p-adic valuation of n , and n is an extravagant number in base b ifK _{b(n)} < \sum _{{\stackrel{p} {p prime }}} K_b(p) + \sum_ K_b(v_p(n)).

See also

• Equidigital number
• Frugal number

References

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

Notes and References

1. Book: Darling, David J. . The universal book of mathematics: from Abracadabra to Zeno's paradoxes . 2004 . . 978-0-471-27047-8 . 102 .