[seqfan] Any other thoughts, opinions on (in)finiteness of A020995?
Alonso Del Arte
alonso.delarte at gmail.com
Mon Mar 19 18:25:52 CET 2012
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}}\right)/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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