Frugal number

A frugal number is a natural number that has more digits than the number of digits in its prime factorization (including exponents).^[1] For example, using base-10 arithmetic, the first few frugal numbers are 125 (5³), 128 (2⁷), 243 (3⁵), and 256 (2⁸). Frugal numbers also exist in other bases; for instance, in binary arithmetic thirty-two is a frugal number, since 10¹⁰¹ = 100000.

The base-10 frugal numbers up to 2000 are:

125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875 (sequence A046759 in the OEIS)

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

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.

References

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

[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 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 Descartes Refactorable Superperfect

See also

Notes

References