cp's OEIS Frontend

A138335 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.

%I A138335 #18 May 27 2025 15:59:52
%S A138335 19,28,29,34,36,37,39,43,50,52,62,68,71,74,75,87,89,94,110,113,128,
%T A138335 129,130,132,137,143,153,169,174,189,201,203,207,209,211,217,240,241,
%U A138335 242,252,253,268,274,275,278,279,284,286,287,297
%N A138335 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 A138335 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 A138335 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 A138335 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.
%C A138335 This sequence is indeed ill defined. One can get the same approximation of Pi to a given precision with infinitely many distinct quadratic polynomials and any such polynomial that gives Pi to n+1 digits also gives Pi to n digits, so this sequence shouldn't have any term. Also, the 18-digit "root" given in the example isn't a root, the polynomial has a value of -5e-13 at this x-value. - _M. F. Hasler_, May 21 2025
%e A138335 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 A138335 << NumberTheory`Recognize`
%t A138335 b = {}; a = {};
%t A138335 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 A138335 dead
%O A138335 1,1
%A A138335 _Artur Jasinski_, Mar 15 2008