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a(n) = n-th decimal digit of the decimal expansion of the n-th Farey fraction ordered by rank.
1

%I #19 Oct 30 2024 08:05:41

%S 0,9,0,3,6,0,0,0,0,0,0,6,3,4,5,5,2,5,8,0,0,0,0,1,2,4,5,7,8,0,0,0,0,0,

%T 1,7,3,5,5,3,7,1,9,3,6,3,6,0,1,3,7,6,3,1,6,9,9,1,7,5,7,5,2,7,7,6,3,6,

%U 6,3,3,6,3,0,0,0,0,0,0,0,0,8,7,4,9,7,1,0,5,7,1,9,1,4,5,5,9,5,7,8,1,2,4,8,6

%N a(n) = n-th decimal digit of the decimal expansion of the n-th Farey fraction ordered by rank.

%C Used as an example in the Schaffter link to support Cantor's diagonal argument. Most probably irrational and possible normal.

%D Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin Heidelberg NY, Section of Analysis, Chptr 15, "Sets, function, and the continuum hypothesis", 2000, pp. 87 - 98.

%D Georg Cantor, Über eine Eigenschaft des Inbegriffes aller reellen Zahlen ("On the Characteristic Property of All Real Numbers")

%D Timothy Gowers, Editor, with June Barrow-Green & Imre Leader, Assc. Editors, The Princeton Companion to Mathematics, Princeton Un. Press, Princeton & Oxford, 2008, pp. 171 & 779

%D Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §7.5 Transfinite Numbers, pp. 257-262.

%H Richard Lipton, <a href="http://rjlipton.wordpress.com/2010/01/20/are-the-reals-really-uncountable/">Gödel's Lost Letter and P=NP</a>

%H Luke Mastin, <a href="http://www.storyofmathematics.com/19th_cantor.html">19th Century Mathematics - Cantor</a>

%H Tom Schaffter, <a href="http://wiki.laptop.org/images/2/28/Cantors_Diagonal_Method.pdf">Cantor's Diagonal Argument: Proof and Paradox</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CantorDiagonalMethod.html">Cantor Diagonal Method</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cantor&#39;s_diagonal_argument">Cantor's diagonal argument</a>

%F The n-th decimal digit of the n-th Farey fraction in order, i.e., 0, 1 (=0.99999...), 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, ..., . (this is A038566/A038567)

%e The first decimal digit of 0 is 0, the second decimal digit of 1 (=0.99999...) is 9, the third decimal digit of 1/2 is 0, the fourth decimal digit of 1/3 is 3, the fifth decimal digit of 2/3 is 6, ..., the twelfth decimal digit of 1/6 is 6, the thirteenth decimal digit of 5/6 is 3, the fourteenth decimaldigit of 1/7 is 4, ..., .

%t FareyOrder[n_] := Select[ Table[a/n, {a, n}], Denominator[#] == n &]; lst = Join[{0, .999999}, Flatten[ Table[ FareyOrder[n], {n, 2, 19}]]]; f[n_] := RealDigits[ lst[[n]], 10, 2 n][[1, n]]; Array[f, 105]

%Y Cf. A071989, A038566, A038567.

%K easy,base,nonn

%O 1,2

%A _Robert G. Wilson v_, Mar 14 2013