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29 maximal partitions, I think it's clear why it fails. We have the partition 29=5+7+8+9, whose LCM is 2520. But, there is no k for which 2520=lcm(k,g(29-k)). For example, if we look at lcm(5,g(24)), it turns out that 24=7+8+9 wasn't a maximal partition for 24, but it's the one that gives the best lcm with 5. I dunno how much of that made sense, but the recursion formula is certainly wrong. I'll just remove it now.
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On an unrelated note, I don't agree with what you've added about g(n) growing like exp(sqrt(n log(n))). It is unclear what you mean by grows like, but log(f(n)) ~ sqrt(n log(n)) does not imply f(n) ~ exp(sqrt(n log(n))), e.g. for f(n) = n exp(sqrt(n log(n))). I do not know whether g(n) ~ exp(sqrt(n
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Mock you? Not at all. Since you found a partition whose LCM is 2520, that means that g(29) really is at least 2520, and the recursion formula, as you note, fails to give that result. I've just been playing with it, and 29 is the first natural number for which it fails. If you write out the first
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Informally speaking, ln(g(n)) ~= sqrt(n ln(n)) +/- epsilon (epsilon ~= 0), so g(n) ~= exp(sqrt(n ln(n))) * exp(epsilon), and since exp(epsilon) ~= 1, g(n) ~= exp(sqrt(n ln(n))). Is something very wrong about this pseudo-proof?
184:; it's just the properties of LCM applied to the definition of the Landau function. If that formula gives g(29)=2310, then g(29)=2310. Maybe Sloane's has the error. What evidence do you have that g(29) is
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323:". The citation in that article is in French, so I am not sure of what it says, although my friend (who is actually a mathematician) believes that it only proves "for sufficiently large
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I think asserting it is straightforward to prove and then using "really" twice is rather condescending, however having a pissing match about this is a waste of our time.
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Goodness, I didn't mean that at all, nor do I want a "pissing match". I apologize. I have no desire to condescend to you. What an unfortunate miscommunication. :-(
327:". (I should not actually say "only"; as the claim is equivalence, changing from "sufficiently large" to "all" strengthens one direction and weakens the other.)
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I disagree. f(n) = n exp(h(n)) does NOT mean log(f(n)) ~ h(n). You're missing a log(n) in there, which can't be removed since it is neither constant nor -: -->
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I disagree with the preceding recursive formula. IMNSHO, it gives the wrong value for g(29) (it gives 2310, not the correct 2520). (comment by
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Feel free to mock me. 2^3 + 3^2 + 5 + 7 = 29, 2^3 * 3^2 * 5 * 7 = 2520.
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Good point. Oops. Unsubstantiated claim removed. Good night.
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2520? An actual partition of 29 whose LCM really is 2520?
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