About the "Hoppa's numbers" (according to Petrov and not only...)

Discussion in 'Number Theory' started by olgaSamara, Aug 24, 2021.

  1. olgaSamara

    olgaSamara

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    The same Petrov has an article "[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), where he talks about the so-called "Hopp's numbers". I immediately warn you that despite the title, the article is positioned as exclusively mathematical. No propaganda of esotericism, etc.

    The "Hopp's numbers" themselves correspond to the statement:

    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.

    For two-digit numbers, it seems that the only example (I'm really not sure about this!) can be the numbers 81 and 54. But in general, there are infinitely many such numbers. In itself, their sequence is not of particular interest. I am not in any way promoting esotericism, but "you can't throw the words out of the song". The fact is that the "Hopp's numbers" come into contact with a certain principle: A/Ω. It dates back to the time of the Pythagoreans. We will not discuss here the magic component of this principle. And just note that the "Hopp's numbers" are interesting in their combination. And this may already be of interest for mathematics, both entertaining and practical (in some cases).

    So, for example, Petrov suggests considering numbers of the form A^Ω and Ω^A. Examines them for divisibility, but operates only with the numbers 81 and 54. Among others, we consider the number 81^54-1, which probably refers to a colossally excessive number (more than 25 million divisors). If there are often Colossally abundant number among such numbers, then this is already a huge interest for programming, for example.

    A little history: who is Hoppa and why are these numbers named after him? Quote from one forum (from syndicatel):

    So, the whole story began at the dawn of the Internet (the 1990s?), when it was customary to communicate via e-mail with university computers. So, a certain student of Filipino origin, M. Hoppa, in one such correspondence on the topic of "entertaining" mathematics, mentioned that there are such interesting numbers that have been known since ancient times. And briefly told about them. I believe that it was with the light hand of Petrov that these numbers began to be called "Hopp's numbers", although this may not be the case.
     
    olgaSamara, Aug 24, 2021
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  2. olgaSamara

    olgaSamara

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    Here's what I thought. For the known numbers 81 and 54, everything is clear:

    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

    since they are two-digit and when raised to the ninth power, they will give up to 18 digits in the resulting number (99^9). But then, for three digits (three-digit) of the original, you will get a maximum of only 27 digits(999^9). The maximum possible sum of digits that makes up a number of 27 digits is 243. But if you check the numbers up to 243^9, it seems that none of them will fall under the approval condition...

    And in general, the function of increasing the sum of the digits of the received number from increasing the digits of the original number will be such that in fact the sum of the digits will always "lag" from the original value of the number. am I right? That is, the statement is true only for two-digit numbers?!
     
    olgaSamara, Aug 25, 2021
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