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A010769 Decimal expansion of 7th root of 2.

%I A010769 #33 Sep 05 2022 16:23:28
%S A010769 1,1,0,4,0,8,9,5,1,3,6,7,3,8,1,2,3,3,7,6,4,9,5,0,5,3,8,7,6,2,3,3,4,4,
%T A010769 7,2,1,3,2,5,3,2,6,6,0,0,7,8,0,1,2,4,1,6,5,5,1,4,5,3,2,4,6,4,1,4,2,1,
%U A010769 0,6,3,2,2,8,8,0,3,8,0,9,8,0,7,1,6,5,9,8,2,8,9,8,8,6,3,0,2,0,0
%N A010769 Decimal expansion of 7th root of 2.
%C A010769 This is also the unique positive attractor of the mapping M(x) = sqrt(sqrt(sqrt(2*x))). In general, (p^N-1)-th root of a number f can be approximated by iterating the mapping M(x) = (f*x)^(1/p^N). The convergence is very fast. In this case, p=2, N=3, and f=2. In the form "evaluate the 3rd (or 7th or 15th) root of a number using only square roots", the insight is usable as a recreational math puzzle. - _Stanislav Sykora_, Oct 26 2015
%H A010769 Vincenzo Librandi, <a href="/A010769/b010769.txt">Table of n, a(n) for n = 1..5000</a>
%F A010769 Equals Product_{k>=0} (1 + (-1)^k/(7*k + 6)). - _Amiram Eldar_, Jul 29 2020
%e A010769 1.104089513673812337649505387623...
%t A010769 RealDigits[N[2^(1/7), 100]][[1]] (* _Vincenzo Librandi_, Apr 02 2013 *)
%t A010769 RealDigits[Surd[2,7],10,120][[1]] (* _Harvey P. Dale_, Sep 05 2022 *)
%o A010769 (PARI) sqrtn(2,7) \\ _Charles R Greathouse IV_, Apr 15 2014
%o A010769 (PARI) { default(realprecision, 100); x= 2^(1/7); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ _Altug Alkan_, Nov 14 2015
%Y A010769 Cube roots (p=2,N=2) for various f: A002580 (2), A002581 (3), A005480 (4), A010582 (10), A092041 (e). 7th roots (p=2,N=3): A246709 (3), A011186 (4), A011201 (5), A011276 (10), A092516 (e). 8th roots (p=3,N=2): A010770 (2), A246710 (3), A011202 (5), A011277 (10). 15th roots (p=2,N=4): A010777(2), A011194(4), A011209(5), A011284(10). - _Stanislav Sykora_, Oct 26 2015
%K A010769 nonn,cons
%O A010769 1,4
%A A010769 _N. J. A. Sloane_