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.

See also

• Equidigital number
• Extravagant number

Notes

References

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

Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number
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 Sociable Betrothed
Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant