Extravagant number
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
- Equidigital number
- Frugal number
Notes
- ^ 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
- R.G.E. Pinch (1998), Economical Numbers.
- Chris Caldwell, The Prime Glossary: extravagant number at The Prime Pages.
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Divisibility-based sets of integers
- Integer factorization
- Divisor
- Unitary divisor
- Divisor function
- Prime factor
- Fundamental theorem of arithmetic
- Prime
- Composite
- Semiprime
- Pronic
- Sphenic
- Square-free
- Powerful
- Perfect power
- Achilles
- Smooth
- Regular
- Rough
- Unusual
- Equidigital
- Extravagant
- Frugal
- Harshad
- Polydivisible
- Smith