Extravagant number

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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 [math]\displaystyle{ b \gt 1 }[/math] be a number base, and let [math]\displaystyle{ K_b(n) = \lfloor \log_{b}{n} \rfloor + 1 }[/math] be the number of digits in a natural number [math]\displaystyle{ n }[/math] for base [math]\displaystyle{ b }[/math]. A natural number [math]\displaystyle{ n }[/math] has the prime factorisation

[math]\displaystyle{ n = \prod_{\stackrel{p \,\mid\, n}{p\text{ prime}}} p^{v_p(n)} }[/math]

where [math]\displaystyle{ v_p(n) }[/math] is the p-adic valuation of [math]\displaystyle{ n }[/math], and [math]\displaystyle{ n }[/math] is an extravagant number in base [math]\displaystyle{ b }[/math] if

[math]\displaystyle{ K_b(n) \lt \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)). }[/math]

See also