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).[1] For example, in base 10, 125 = 5³, 128 = 2⁷, 243 = 3⁵, and 256 = 2⁸ are frugal numbers (sequence A046759 in the OEIS). 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 about a frugal number, but also about a number which is either frugal or equidigital.

Mathematical definition

Let ${\displaystyle b>1}$ be a number base, and let ${\displaystyle K_{b}(n)=\lfloor \log _{b}{n}\rfloor +1}$ be the number of digits in a natural number ${\displaystyle n}$ for base ${\displaystyle b}$. A natural number ${\displaystyle n}$ has the integer factorisation

${\displaystyle n=\prod _{\stackrel {p\mid n}{p{\text{ prime}}}}p^{v_{p}(n)}}$

and is an frugal number in base ${\displaystyle b}$ if

${\displaystyle K_{b}(n)>\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}K_{b}(p)+\sum _{\stackrel {p^{2}\mid n}{p{\text{ prime}}}}K_{b}(v_{p}(n))}$

where ${\displaystyle v_{p}(n)}$ is the p-adic valuation of ${\displaystyle n}$.

See also

• Equidigital number
• Extravagant number

Notes

1. Darling, David J. (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley & Sons. p. 102. ISBN 978-0-471-27047-8.

References

• R.G.E. Pinch (1998), Economical Numbers