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%I #27 Mar 01 2023 04:42:00
%S 4,3,6,4,5,7,6,9,5,4,7,7,6,8,3,3,0,4,5,9,8,3,3,0,8,9,6,2,4,1,0,4,4,0,
%T 6,7,9,5,1,7,4,1,3,5,7,5,7,7,4,8,8,9,5,0,5,0,6,5,1,7,3,3,9,9,7,7,6,1,
%U 4,9,9,2,7,8,5,8,4,9,4,5,4,2,8,0,2,1,7,7,4,7,4,8,1,8,4,5,7,5,8,0,0,0,1,4,3
%N a(n) = n-th decimal digit of the fractional part of the square root of the n-th nonsquare number (A000037).
%C Regarded as a decimal fraction, 0.43645769547768330... is likely to be an irrational number.
%D Martin Aigner & 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 Chai Wah Wu, <a href="/A071989/b071989.txt">Table of n, a(n) for n = 1..10000</a>
%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's_diagonal_argument">Cantor's diagonal argument</a>.
%F The n-th nonsquare number is m(n) = n + ( sqrt( n + sqrt(n)).
%e Sqrt(2)=1.4142135... -> the 1st decimal digit is 4,
%e sqrt(3)=1.7320508... -> the 2nd decimal digit is 3,
%e sqrt(5)=2.2360679... -> the 3rd decimal digit is 6,
%e sqrt(6)=2.4494897... -> the 4th decimal digit is 4, etc.
%t q[n_] := (m = Floor[n + Sqrt[n + Sqrt[n]]]; Floor[ Mod[ 10^n*Sqrt[m], 10]]); Table[ q[n], {n, 1, 105}]
%Y Cf. A000037, A071901, A216251.
%K base,nonn
%O 1,1
%A _Robert G. Wilson v_, Jun 17 2002
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