login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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

%I #40 Mar 31 2024 12:05:05

%S 1,7,14,113,226,339,452,565,678,791,904,1017,1130,1243,33215,99532,

%T 364913,1725033,3450066,25510582,131002976,340262731,811528438,

%U 1963319607,6701487259,13402974518,20104461777,26805949036,33507436295,40208923554,567663097408

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

%C 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.

%H AMS, <a href="http://www.ams.org/publicoutreach/feature-column/fcarc-irrational2">Rational approximation of irrational numbers</a>

%H Jon E. Schoenfield, <a href="/A346534/a346534.txt">Magma program with explanation of algorithm</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)">Hurwitz's theorem (number theory)</a>

%e 22/7 ~ 3.1428571 and E/M ~ 0.1385.

%e 355/113 ~ 3.1415929 and E/M ~ 0.0076.

%e From _Jon E. Schoenfield_, Aug 06 2021: (Start)

%e k j E = |j/k - Pi| M = 1/(sqrt(5)*k^2) E/M

%e ----- ------ -------------- ------------------- -------

%e 1 3 0.141592653590 0.44721359549995794 0.31661

%e 7 22 0.001264489267 0.00912680807142771 0.13855

%e 14 44 0.001264489267 0.00228170201785693 0.55419

%e 113 355 0.000000266764 0.00003502338440755 0.00762

%e 226 710 0.000000266764 0.00000875584610189 0.03047

%e 339 1065 0.000000266764 0.00000389148715639 0.06855

%e 452 1420 0.000000266764 0.00000218896152547 0.12187

%e 565 1775 0.000000266764 0.00000140093537630 0.19042

%e 678 2130 0.000000266764 0.00000097287178910 0.27420

%e 791 2485 0.000000266764 0.00000071476294709 0.37322

%e 904 2840 0.000000266764 0.00000054724038137 0.48747

%e 1017 3195 0.000000266764 0.00000043238746182 0.61696

%e 1130 3550 0.000000266764 0.00000035023384408 0.76167

%e 1243 3905 0.000000266764 0.00000028944945791 0.92163

%e 33215 104348 0.000000000332 0.00000000040536522 0.81810

%e (End)

%t 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 *)

%o (Magma) // See Links.

%o (PARI) is(k) = my(j=round(Pi*k)); abs(j/k - Pi)*sqrt(5)*k^2 < 1; \\ _Jinyuan Wang_, Aug 06 2021

%Y Cf. A000796, A001203, A063673.

%Y Cf. A002163 (sqrt(5)).

%K nonn,frac

%O 1,2

%A _June Richardson_, Jul 22 2021

%E a(17)-a(19) from _Jinyuan Wang_, Aug 06 2021

%E a(20)-a(31) from _Jon E. Schoenfield_, Aug 06 2021