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

Let be a number base, and let be the number of digits in a natural number for base . A natural number has the prime factorisation

where is the p-adic valuation of , and is an extravagant number in base if

See also


  1. ^ 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.