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

%I A346534 #40 Mar 31 2024 12:05:05
%S A346534 1,7,14,113,226,339,452,565,678,791,904,1017,1130,1243,33215,99532,
%T A346534 364913,1725033,3450066,25510582,131002976,340262731,811528438,
%U A346534 1963319607,6701487259,13402974518,20104461777,26805949036,33507436295,40208923554,567663097408
%N A346534 Denominators of approximations j/k for Pi such that abs(j/k - Pi)*sqrt(5)*k^2 < 1.
%C A346534 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 A346534 AMS, <a href="http://www.ams.org/publicoutreach/feature-column/fcarc-irrational2">Rational approximation of irrational numbers</a>
%H A346534 Jon E. Schoenfield, <a href="/A346534/a346534.txt">Magma program with explanation of algorithm</a>
%H A346534 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(number_theory)">Hurwitz's theorem (number theory)</a>
%e A346534 22/7 ~ 3.1428571 and E/M ~ 0.1385.
%e A346534 355/113 ~ 3.1415929 and E/M ~ 0.0076.
%e A346534 From _Jon E. Schoenfield_, Aug 06 2021: (Start)
%e A346534     k       j    E = |j/k - Pi|  M = 1/(sqrt(5)*k^2)    E/M
%e A346534   -----  ------  --------------  -------------------  -------
%e A346534       1       3  0.141592653590  0.44721359549995794  0.31661
%e A346534       7      22  0.001264489267  0.00912680807142771  0.13855
%e A346534      14      44  0.001264489267  0.00228170201785693  0.55419
%e A346534     113     355  0.000000266764  0.00003502338440755  0.00762
%e A346534     226     710  0.000000266764  0.00000875584610189  0.03047
%e A346534     339    1065  0.000000266764  0.00000389148715639  0.06855
%e A346534     452    1420  0.000000266764  0.00000218896152547  0.12187
%e A346534     565    1775  0.000000266764  0.00000140093537630  0.19042
%e A346534     678    2130  0.000000266764  0.00000097287178910  0.27420
%e A346534     791    2485  0.000000266764  0.00000071476294709  0.37322
%e A346534     904    2840  0.000000266764  0.00000054724038137  0.48747
%e A346534    1017    3195  0.000000266764  0.00000043238746182  0.61696
%e A346534    1130    3550  0.000000266764  0.00000035023384408  0.76167
%e A346534    1243    3905  0.000000266764  0.00000028944945791  0.92163
%e A346534   33215  104348  0.000000000332  0.00000000040536522  0.81810
%e A346534 (End)
%t A346534 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 A346534 (Magma) // See Links.
%o A346534 (PARI) is(k) = my(j=round(Pi*k)); abs(j/k - Pi)*sqrt(5)*k^2 < 1; \\ _Jinyuan Wang_, Aug 06 2021
%Y A346534 Cf. A000796, A001203, A063673.
%Y A346534 Cf. A002163 (sqrt(5)).
%K A346534 nonn,frac
%O A346534 1,2
%A A346534 _June Richardson_, Jul 22 2021
%E A346534 a(17)-a(19) from _Jinyuan Wang_, Aug 06 2021
%E A346534 a(20)-a(31) from _Jon E. Schoenfield_, Aug 06 2021