frugal number

A frugal number or economical number n is an integer with a base b representationPlanetmathPlanetmath of k digits for which the prime factorizationMathworldPlanetmath uses fewer than k digits (with repeated prime factorsMathworldPlanetmath grouped with exponents and the digits of those exponents counted whenever greater than 1). The first few frugal numbers in base 10 are 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875, etc. (listed in A046759 of Sloane’s OEIS). For example, 128 is frugal in base 10 because it is written with three digits, while its factorization of 27 uses just two digits. 128 also happens to be frugal in binary. If we regard 1 as not prime, then 1 is a frugal number in all positional bases (for example, Mathematica returns its factorization as an empty set).


  • 1 D. Darling, “Economical number” in The Universal Book of Mathematics: From Abracadabra To Zeno’s paradoxes. Hoboken, New Jersey: Wiley (2004)
Title frugal number
Canonical name FrugalNumber
Date of creation 2013-03-22 16:41:15
Last modified on 2013-03-22 16:41:15
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A63
Synonym economical number
Related topic EquidigitalNumber
Related topic ExtravagantNumber