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). For example, in base 10, 125 = 5<sup>3</sup>, 128 = 2<sup>7</sup>, 243 = 3<sup>5</sup>, and 256 = 2<sup>8</sup> are frugal numbers . The first frugal number which is not a prime power is 1029 = 3 ÃÂ 7<sup>3</sup>. In base 2, thirty-two is a frugal number, since 32 = 2<sup>5</sup> is written in base 2 as 100000 = 10<sup>101</sup>.
The term economical number has been used for a frugal number, but also for a number which is either frugal or equidigital.
Let be a number base, and let be the number of digits in a natural number for base . A natural number has the prime factorisation
where is the p-adic valuation of , and is an frugal number in base if