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Extravagant number
Number that has fewer digits than the number of digits in its prime factorization
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). 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 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)
Notes
References
- R.G.E. Pinch (1998), Economical Numbers.
- Chris Caldwell, The Prime Glossary: extravagant number at The Prime Pages.
References
- Darling, David J.. (2004). "The universal book of mathematics: from Abracadabra to Zeno's paradoxes". [[John Wiley & Sons]].
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