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
Examples
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)
Links
- AMS, Rational approximation of irrational numbers
- Jon E. Schoenfield, Magma program with explanation of algorithm
- Wikipedia, Hurwitz's theorem (number theory)
Programs
-
Magma
// See Links.
-
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 *) -
PARI
is(k) = my(j=round(Pi*k)); abs(j/k - Pi)*sqrt(5)*k^2 < 1; \\ Jinyuan Wang, Aug 06 2021
Extensions
a(17)-a(19) from Jinyuan Wang, Aug 06 2021
a(20)-a(31) from Jon E. Schoenfield, Aug 06 2021
Comments