

A216251


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


1



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
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OFFSET

1,2


COMMENTS

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


REFERENCES

Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, Second Edition, SpringerVerlag, 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 BarrowGreen & 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. 257262.


LINKS

Table of n, a(n) for n=1..105.
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


FORMULA

The nth decimal digit of the nth 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)


EXAMPLE

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, ..., .


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]


CROSSREFS

Cf. A071989, A038566, A038567.
Sequence in context: A062523 A093961 A154903 * A011110 A211884 A154464
Adjacent sequences: A216248 A216249 A216250 * A216252 A216253 A216254


KEYWORD

easy,base,nonn


AUTHOR

Robert G. Wilson v, Mar 14 2013


STATUS

approved



