Frugal number explained

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 . 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 for a frugal number, but also for a number which is either frugal or equidigital.

Mathematical definition

Let

b > 1 be a number base, and let K _{b(n)} = \lfloor log _{b{n}} \rfloor + 1 be the number of digits in a natural number n for base b . A natural number n has the prime factorisationn = \prod _{\stackrel{p} {p prime }} p^where v _{p(n)} is the p-adic valuation of n , and n is an frugal number in base b ifK _{b(n)} > \sum _{{\stackrel{p} {p prime }}} K_b(p) + \sum_ K_b(v_p(n)).

See also

• Equidigital number
• Extravagant number

References

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

Notes and References

1. Book: Darling, David J. . The universal book of mathematics: from Abracadabra to Zeno's paradoxes . 2004 . . 978-0-471-27047-8 . 102 .