A216251 a(n) = n-th decimal digit of the decimal expansion of the n-th Farey fraction ordered by rank.
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, 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, 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
Offset: 1
Examples
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, ..., .
References
- 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.
- Georg Cantor, Über eine Eigenschaft des Inbegriffes aller reellen Zahlen ("On the Characteristic Property of All Real Numbers")
- 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
- Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §7.5 Transfinite Numbers, pp. 257-262.
Links
- Richard Lipton, Gödel's Lost Letter and P=NP
- Luke Mastin, 19th Century Mathematics - Cantor
- Tom Schaffter, Cantor's Diagonal Argument: Proof and Paradox
- Eric Weisstein's World of Mathematics, Cantor Diagonal Method
- Wikipedia, Cantor's diagonal argument
Programs
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Mathematica
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]
Comments