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A216251 a(n) = n-th decimal digit of the decimal expansion of the n-th 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 (list; graph; refs; listen; history; text; internal format)
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, 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

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 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)

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

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Last modified January 19 21:47 EST 2020. Contains 331066 sequences. (Running on oeis4.)