In order to find the next element I consulted The on-line encyclopedia of integer sequences. Funnily enough they only have the sequence up to 82000. There is a reference to this paper where the author claims that the next number would be at least 10**21 according to his experiments.
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I can atest to it being greater than 95367431640625.
Even assuming that my "sum of single powers" hypothesis is correct; I think it will take more cpu power than I'm prepared to expend to look much beyond that.
With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
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Actually, it says at least 102184, as I previously mentioned.
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82000 is a sum of powers of all the inputs
That's sort of by definition: if the number is expressed with only 1 and 0 in a given base, then it will be sum of simple powers of that base.
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Hm. A phraseology problem I think.
Below are all the numbers less than 3^11, that when encoded base-3 use only 0 1:
All of them can be defined as the sums of multiples of powers of 3. But only the first number in each block is a power of 3.
And 82000 is a sum of a selection of those first numbers in each block. And that is so for 4 & 5 also.
And, if it holds true for the higher numbers in the sequence (and they are going to be very large) then not having to consider all the other numbers in each of those blocks is a significant saving.
So worth pointing out don't you think? Even if I need to clarify the meaning or use better phraseology.
How about a sum of single powers? Or discrete powers?
With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
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[d/l] |
Yes, it is probably a phraseology problem or a misunderstanding, but I still don't agree with this sentence:
All of them can be defined as the sums of multiples of powers of 3. But only the first number in each block is a power of 3.
To me, these numbers are all sums of single powers of 3. For example, taking the beginning of your list:
1 -- 3**0
3 4 -- 3**1, 3**0 + 3**1
9 10 12 13 -- 3**2, 3**0 + 3**2, 3**1 + 3**2, 3**0 + 3**1 + 3**2
etc.
So they are all sums of pure (or single) powers of 3, not sums of multiples of powers of 3 (which would imply numbers expressed with other digits than 0 and 1 in base 3). And so is 82000.
And I agree with you that you don't have to consider these other numbers, only those that are pure powers of 3 are of interest for the search; so, as you said, only the first one of each block if you want to figure out whether 82000 or any other number qualifies the test.
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[d/l] |
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