[seqfan] Re: Any other thoughts, opinions on (in)finiteness of A020995?

[David Wilson]

When one looks at a graph of "Goldbach's comet", one "sees" the extreme 
unlikelihood that the comet will ever touch the x-axis, providing a 
counterexample to Goldbach's conjecture.

Perhaps a similar plot would give one a feeling for the likelihood that 
A020995 is complete. I suggest plotting

     Plot A: y = digitsum(fib(x))
     Plot B: Line y = cx where c = 4.5 log_10(phi) = .9404443811...
     Plot C: y = x

for, say, 0 <= x <= 10000.

I suspect that Plot A will appear as a "Fibonacci digit sum comet" 
hovering around line B, and that line C will eventually wander far 
enough away from the comet's "tail" that one would be extremely 
confident that there will be no more collisions.

On 3/19/2012 8:11 PM, Sven Simon wrote:
> Hello,
>
> these numbers were checked up to large indices, and no further numbers with
> the property were found. They were checked by David Terr for his article
> (Link in A020995). He made a statistical analysis and predicted 684+-26
> numbers for the same property in base 11. In fact I found 710 of these
> numbers in base 11 (A025490). Lucas numbers have the same size, but there
> were only 284 having the property in base 11 (A025491). So you see, that
> Terr's analysis was quite accurate. This difference is explained a little -
> but not in detail - in Terr's article. With Lucas numbers the sequence ends
> 948539,973261,983101,2528952, so there happened a statistical runaway (right
> English?). But these numbers were checked up to indices 7500000, so there is
> nearly no chance for more numbers even in base 11. And the same is true for
> Fibonacci in base 10, where T.D. Noe checked up to large indices (A004090).
> There is no prove for it, but you won't find a new one in base 10 - I would
> bet. No risk no fun.
> Sven
>
> -----Ursprüngliche Nachricht-----
> Von: seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu] Im
> Auftrag von Alonso Del Arte
> Gesendet: Montag, 19. März 2012 18:26
> An: Sequence Fanatics Discussion list
> Betreff: [seqfan] Any other thoughts, opinions on (in)finiteness of A020995?
>
> Long ago, Robert G. Wilson v suggested that the sequence of n such that the
> sum of the base 10 digits of Fibonacci(n) is equal to n (A020995) might be
> infinite, though only twenty terms are known and the largest known term is
> quite small. In 2006, Stefan quoted an argument from Robert Dawson that not
> only is the sequence likely finite, we might already know all the terms.
>
> Most recently, Charles posted the following argument in a comment for the
> Sequence of the Day for April 20, saying that it needs to be checked:
>
> "The number of digits in the n-th Fibonacci number is
> <math>n\log_{10}\varphi+o(1)</math>, so the expected digit sum is about
> 0.94n.  Modeling the result as a normal distribution, the variance is about
> <math>82.5\log_{10}\varphi</math>  and so the heuristic probability that the
> digits of<math>F_n</math>  are large enough to be in the sequence is about
> <math>\operatorname{erfc}\left(n\frac{1-4.5\log_{10}\varphi}{\sqrt{165}}\rig
> ht)/2.</math>
> This decays rapidly: by n = 10,000 it is below<math>10^{-935}.</math>"
>
> Have you any other thoughts, opinions, those of you who have pondered this
> or similar questions?
>
> Al
>
> --
> Alonso del Arte
> Author at
> SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte>
> Musician at ReverbNation.com<http://www.reverbnation.com/alonsodelarte>
>
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