A346534 Denominators of approximations j/k for Pi such that abs(j/k - Pi)*sqrt(5)*k^2 < 1.

1, 7, 14, 113, 226, 339, 452, 565, 678, 791, 904, 1017, 1130, 1243, 33215, 99532, 364913, 1725033, 3450066, 25510582, 131002976, 340262731, 811528438, 1963319607, 6701487259, 13402974518, 20104461777, 26805949036, 33507436295, 40208923554, 567663097408

Offset

1,2

Comments

Define two parameters E and M for a rational approximation j/k for an irrational number x: E = abs(j/k - x) (the absolute error) and M = 1/(sqrt(5)*k^2). Hurwitz's theorem states that every real number has infinitely many rational approximations that satisfy E/M < 1, making each such approximation a "strong approximation". This sequence lists the denominators of such numbers for the irrational number Pi.

Links

Table of n, a(n) for n=1..31.

AMS, Rational approximation of irrational numbers

Jon E. Schoenfield, Magma program with explanation of algorithm

Wikipedia, Hurwitz's theorem (number theory)

Example

22/7 ~ 3.1428571 and E/M ~ 0.1385.
355/113 ~ 3.1415929 and E/M ~ 0.0076.
From Jon E. Schoenfield, Aug 06 2021: (Start)
    k       j    E = |j/k - Pi|  M = 1/(sqrt(5)*k^2)    E/M
  -----  ------  --------------  -------------------  -------
      1       3  0.141592653590  0.44721359549995794  0.31661
      7      22  0.001264489267  0.00912680807142771  0.13855
     14      44  0.001264489267  0.00228170201785693  0.55419
    113     355  0.000000266764  0.00003502338440755  0.00762
    226     710  0.000000266764  0.00000875584610189  0.03047
    339    1065  0.000000266764  0.00000389148715639  0.06855
    452    1420  0.000000266764  0.00000218896152547  0.12187
    565    1775  0.000000266764  0.00000140093537630  0.19042
    678    2130  0.000000266764  0.00000097287178910  0.27420
    791    2485  0.000000266764  0.00000071476294709  0.37322
    904    2840  0.000000266764  0.00000054724038137  0.48747
   1017    3195  0.000000266764  0.00000043238746182  0.61696
   1130    3550  0.000000266764  0.00000035023384408  0.76167
   1243    3905  0.000000266764  0.00000028944945791  0.92163
  33215  104348  0.000000000332  0.00000000040536522  0.81810
(End)

Mathematica

a={}; For[k=1, k<=10^6, k++, If[Abs[Round[k Pi]/k-Pi]Sqrt[5] k^2<1, AppendTo[a, k]]]; a (* Stefano Spezia, Aug 07 2021 *)

Prog

(Magma) // See Links.
(PARI) is(k) = my(j=round(Pi*k)); abs(j/k - Pi)*sqrt(5)*k^2 < 1; \\ Jinyuan Wang, Aug 06 2021

Crossrefs

Cf. A000796, A001203, A063673.

Cf. A002163 (sqrt(5)).

Sequence in context: A143682 A369562 A080451 * A369122 A061522 A213151

Adjacent sequences: A346531 A346532 A346533 * A346535 A346536 A346537

Keyword

nonn,frac

Author

June Richardson, Jul 22 2021

Extensions

a(17)-a(19) from Jinyuan Wang, Aug 06 2021

a(20)-a(31) from Jon E. Schoenfield, Aug 06 2021

Status

approved