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A071989
a(n) = n-th decimal digit of the fractional part of the square root of the n-th nonsquare number (A000037).
2
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, 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, 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
OFFSET
1,1
COMMENTS
Regarded as a decimal fraction, 0.43645769547768330... is likely to be an irrational number.
REFERENCES
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.
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.
FORMULA
a(n) = floor(sqrt(A000037(n))*10^n) mod 10. - Jason Yuen, Aug 20 2024
EXAMPLE
Sqrt(2)=1.4142135... -> the 1st decimal digit is 4,
sqrt(3)=1.7320508... -> the 2nd decimal digit is 3,
sqrt(5)=2.2360679... -> the 3rd decimal digit is 6,
sqrt(6)=2.4494897... -> the 4th decimal digit is 4, etc.
MATHEMATICA
q[n_] := (m = Floor[n + Sqrt[n + Sqrt[n]]]; Floor[ Mod[ 10^n*Sqrt[m], 10]]); Table[ q[n], {n, 1, 105}]
PROG
(Python)
from math import isqrt
def A071989(n): return isqrt(10**(n<<1)*(n+(k:=isqrt(n))+int(n>=k*(k+1)+1)))%10 # Chai Wah Wu, Jul 20 2024
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Robert G. Wilson v, Jun 17 2002
STATUS
approved