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, no matter what base is used.[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 integer factorisation

and is an extravagant number in base if

where is the p-adic valuation of .

See also

Notes

References

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