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). 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 [math]\displaystyle{ b \gt 1 }[/math] be a number base, and let [math]\displaystyle{ K_b(n) = \lfloor \log_b{n} \rfloor + 1 }[/math] be the number of digits in a natural number [math]\displaystyle{ n }[/math] for base [math]\displaystyle{ b }[/math]. A natural number [math]\displaystyle{ n }[/math] has the prime factorisation

[math]\displaystyle{ n = \prod_{\stackrel{p \,\mid\, n}{p\text{ prime}}} p^{v_p(n)} }[/math]

where [math]\displaystyle{ v_p(n) }[/math] is the p-adic valuation of [math]\displaystyle{ n }[/math], and [math]\displaystyle{ n }[/math] is an frugal number in base [math]\displaystyle{ b }[/math] if

[math]\displaystyle{ K_b(n) \gt \sum_{{\stackrel{p \,\mid\, n}{p\text{ prime}}}} K_b(p) + \sum_{{\stackrel{p^2 \,\mid\, n}{p\text{ prime}}}} K_b(v_p(n)). }[/math]

See also