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Are there any Hoppa's numbers other than 81 and 54?

Wed 25 Aug, 2021 02:03 am
There are several formulations of the definitions of "Hoppa's numbers", but it seems to me the most successful according to Petrov "[Petrov I. B. "Numerical study of the divisibility of the "golden numbers of luck: A/Ω = 81/54", SI, 74 p. - 2021 [18+]] (only in Russian):

there are natural multi-valued numbers, such that when raised to a power equal to nine, they generate numbers, the sum of the digits of each of which is equal to the original number.

There are two known numbers 81 and 54, for which:

81^9=150094635296999121, 1+5+0+0+9+4+6+3+5+2+9+6+9+9+9+1+2+1=81

54^9=3904305912313344, 3+9+0+4+3+0+5+9+1+2+3+1+3+3+4+4=54

For three characters: 999^9 will give a number of 27 digits. Even if it were only the digits 9, then 9*27=243. That is, we can get the maximum number 243 for three-digit numbers. But until this number (243^9), we will not get one that falls under the statement.

I think that in general, the function of increasing the sum of the digits of the resulting number from increasing the digits of the original number will be such that in fact the sum of the digits will always be "late" from the original value of the number. Am I right?
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