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 = 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.
The term economical number has been used about a frugal number, but also about 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 integer factorisation
and is an frugal number in base if
where is the p-adic valuation of .