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Frugal number

Number that has more digits than the number of digits in its prime factorization

Summary

Number that has more digits than the number of digits in its prime factorization

In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers . The first frugal number which is not a prime power is 1029 = 3 × 73. In base 2, thirty-two is a frugal number, since 32 = 25 is written in base 2 as 100000 = 10101.

The term economical number has been used for a frugal number, but also for a number which is either frugal or equidigital.

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 frugal number in base b if : K_b(n) \sum_}} K_b(p) + \sum_}} K_b(v_p(n)).

Notes

References

References

  1. Darling, David J.. (2004). "The universal book of mathematics: from Abracadabra to Zeno's paradoxes". [[John Wiley & Sons]].
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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