Frugal number

Last updated December 13, 2024

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.

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 _{\stackrel {p\,\mid \,n}{p{\text{ prime}}}}K_{b}(p)+\sum _{\stackrel {p^{2}\,\mid \,n}{p{\text{ prime}}}}K_{b}(v_{p}(n)).

Notes

↑ 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.

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References

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

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[Darling102-1] 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.

[1]

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

Frugal number

Contents

Mathematical definition

See also

Notes

Related Research Articles

References