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). 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

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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

 

See also

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Notes

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  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.

References

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