Frugal Number

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 $K_b(n) = \lfloor \log_b \rfloor + 1$ be the number of digits in a natural number $n$ for base $b$. A natural number $n$ has the prime factorisation
: $n = \prod_ p^$
where $v_p(n)$ is the ''p''-adic valuation of $n$, and $n$ is an frugal number in base $b$ if
: $K_b(n) > \sum_ K_b(p) + \sum_ K_b(v_p(n)).$

See also

* Equidigital number
* Extravagant number

Notes

References

* R.G.E. Pinch (1998)
Economical Numbers

{{Classes of natural numbers

Base-dependent integer sequences