economical number

Suppose we take a positive integer n and then find its prime-power factorization.  How do the number of digits in n compare to the total number of digits used to write its factorization? For example, 128=27, so the factorization requires fewer; and 30=2.3.5, so this time the factorization requires more.  Santos and Pinch have made the following definitions:

n is if the factorization of n requires examples number with
2 ≤ n ≤ 500,000,000
frugal fewer digits than n does 2k for k > 6 1,455,952
(or equadigital)
the same number of digits as n does all prime numbers86,441,875
extravagant more digits than n does30, 42, 70 and
p# for p > 3
Additionally, a number is said to be economical if it is not extravagant (that is, its prime power factorization requires no more digits than it does).

The examples above show that there are infinitely many of each of these kinds of numbers, but are there arbitrarily long sequences of consecutive ones?  For example, there are strings of consecutive economical numbers of length seven starting at each of 157, 108749, 109997, 121981 and 143421; and a string of length nine starting at 1034429177995381247. However, when Pinch checked up to 1000000, the longest string of consecutive frugal numbers was just two (for example, 4374 and 4375).  Still, Pinch was able to prove that if Dickson’s conjecture holds, then there are infinitely long strings of consecutive frugal (hence also economical) integers!

See Also: SmithNumber, PerfectNumber, DeficientNumber


R. Pinch, "Economical numbers," (1998) preprint. (Annotation available)
B. Santos, "Problem 2204: equidigital representations," J. Recreational Math., 21:1 (1995) 58-59. [Are there arbitrarily long sequence of economical numbers? See also [Pinch98].]
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