Innumber theory, afrugal number is anatural number in a givennumber base that has moredigits than the number of digits in itsprime factorization in the given number base (including exponents).^[1] For example, inbase 10, 125 = 5³, 128 = 2⁷, 243 = 3⁵, and 256 = 2⁸ are frugal numbers (sequenceA046759 in theOEIS). The first frugal number which is not aprime power is 1029 = 3 × 7³. Inbase 2, thirty-two is a frugal number, since 32 = 2⁵ is written in base 2 as 100000 = 10¹⁰¹.

The termeconomical number has been used for a frugal number, but also for a number which is either frugal orequidigital.

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 prime factorisation

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

where${\displaystyle v_p(n)}$ is thep-adic valuation of${\displaystyle n}$, and${\displaystyle n}$ is anfrugal number in base${\displaystyle b}$ if

${\displaystyle K_b(n)>\sum _{\stackrel{p\mid n}{p\text{ prime}}}{K}_b(p)+\sum _{\stackrel{p^2\mid n}{p\text{ prime}}}{K}_b(v_p(n)).}$

• Equidigital number
• Extravagant number

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

• Base-dependent integer sequences

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• Short description is different from Wikidata