Frugal number

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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).[1] For example, in base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers (sequence A046759 in the OEIS), and in base 2, thirty-two is a frugal number, since 100000 = 10101.

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

Mathematical definition[edit]

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 .

See also[edit]


  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.