A138335 - OEIS

%I #11 Dec 09 2017 19:42:42

%S 19,28,29,34,36,37,39,43,50,52,62,68,71,74,75,87,89,94,110,113,128,

%T 129,130,132,137,143,153,169,174,189,201,203,207,209,211,217,240,241,

%U 242,252,253,268,274,275,278,279,284,286,287,297

%N Positions of digits after decimal point in decimal expansion of Pi where the approximation to Pi by a root of a quadratic polynomial does not improve the accuracy.

%C If there is a set of consecutive integers in this sequence starting at k, this means that k-1 is a good approximation to Pi.

%C If the set of successive integers is longer that approximation k-1 better (see A138336). [Sentence is not clear - _N. J. A. Sloane_, Dec 09 2017]

%C Comment from _Joerg Arndt_, Mar 17 2008: Does Mathematica's N[((quantity)), n] round a number (if so, to what base?) or truncate it? Is Mathematica's Recognize[] guaranteed to give the correct relation? I do not think so: that would be a major breakthrough. That is, this sequence may not even be well-defined.

%e a(1)=19 because 3.141592653589793238 (18 digits) is root of -3061495 + 674903*x + 95366*x^2 and 3.1415926535897932385 (19 digits) also is root of that same polynomial -3061495 + 674903*x + 95366*x^2.

%t << NumberTheory`Recognize`

%t b = {}; a = {};

%t Do[k = Recognize[N[Pi,n], 2, x]; If[MemberQ[a, k], AppendTo[b, n], AppendTo[a, k]], {n, 2, 300}]; b (* Artur Jasinski *)

%K nonn,base,less

%O 1,1

%A _Artur Jasinski_, Mar 15 2008