A071989 a(n) = n-th decimal digit of the fractional part of the square root of the n-th nonsquare number (A000037).

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.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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

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

Cf. A000037, A071901, A216251.

Sequence in context: A228780 A333328 A109339 * A290278 A300894 A328258

Adjacent sequences: A071986 A071987 A071988 * A071990 A071991 A071992

Keyword

base,nonn

Author

Robert G. Wilson v, Jun 17 2002

Status

approved