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

1,1

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.

Links

Table of n, a(n) for n=1..50.

Example

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.

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 *)

Crossrefs

Sequence in context: A141417 A264834 A069529 * A304367 A298638 A291884

Adjacent sequences: A138332 A138333 A138334 * A138336 A138337 A138338

Keyword

nonn,base,less

Author

Artur Jasinski, Mar 15 2008

Status

approved