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economical number

A number that has no more digits than there are digits in its prime factorization (including powers). If a number has fewer digits than are in its prime factorization it is known as a frugal number. The smallest frugal is 125, which has three digits, but can be written as 53, which has only two. The next few frugals are 128 (27), 243 (35), 256 (28), 343 (73), 512 (29), 625 (54), and 729 (36). An equidigital number is an economical number that has the same number of digits as make up its prime factorization. The smallest equidigitals are 1, 2, 3, 5, 7, and 10 (= 2 × 5). All prime numbers are equidigital. An extravagant number is one that has fewer digits than are in its prime factorization. The smallest extravagant number is 4 (= 22), followed by 6, 8, and 9. There are infinitely many of each of these kinds of numbers. Are there also arbitrarily long sequences of consecutive ones? Strings of consecutive economical numbers of length seven start at each of 157, 108749, 109997, 121981 and 143421. On the other hand, the longest string of consecutive frugal numbers up to 1000000 is just two (for example, 4374 and 4375). Even so, it has been proved that, if a certain conjecture about prime numbers, known as Dickson's Conjecture is true, then there are arbitrarily long strings of frugals.

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