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.

19, 28, 29, 34, 36, 37, 39, 43, 50, 52, 62, 68, 71, 74, 75, 87, 89, 94, 110, 113, 128, 129, 130, 132, 137, 143, 153, 169, 174, 189, 201, 203, 207, 209, 211, 217, 240, 241, 242, 252, 253, 268, 274, 275, 278, 279, 284, 286, 287, 297

Offset: 1

Author

Artur Jasinski, Mar 15 2008

Keywords

• dead

Comments

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.

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]

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.

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

Examples

			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.
		

Programs

Mathematica

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