Proof needed

[Artur]

Dear Seqfans
Who have idea how to prove:
Some numbers are giving empty set in this procedure. These numebrs are e.g.
Golden Ratio=(1+Sqrt[5])/2, Sqrt[2] what mean that if you need improove accuracy one decimal digit we have to uses new fraction with bigger integers as numerator and denominator.

Method bellow

Any comment appreciated
BEST WISHES
ARTUR



*************************************************************************************
%I A138373
%S A138373 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28, 31, 33, 36, 38, 41, 43, 46, 48, 51, 
53, 56, 61, 63, 66, 68, 71, 73, 76, 81, 83, 86, 88, 91, 96, 101, 106, 111, 
116, 119, 121, 124, 126, 129, 131, 134, 136, 139, 141, 144, 146, 149, 151, 
154, 159, 161, 164, 166, 169, 171, 174, 176, 179, 181, 184, 186, 189, 191, 
194, 196, 199, 201, 204, 206, 209, 211, 214, 216, 219, 221, 224, 229, 231, 
234, 236, 239, 244, 249, 254, 259, 264, 266, 267, 269, 274, 279, 284, 287, 
289, 294, 297, 302, 307, 309, 312, 314, 317, 322, 324, 327, 329, 332, 334, 
337, 339, 342, 344, 347, 349, 352, 354, 357, 359, 362, 364, 367, 372, 374, 
377, 379, 382, 384, 387, 389, 392, 394, 397, 399, 402, 407, 412, 417, 419, 
422, 425, 427, 429, 432, 435, 437, 440, 442, 447, 452, 455, 457, 460, 462, 
465, 467, 470, 472, 475, 477, 480, 482, 485, 487, 490, 492, 495, 497, 500
%N A138373 Positions of digits after decimal point of number Sqrt[5]/2 where the approximation to this number by rational fraction does not improve the accuracy.
%C A138373 Some numbers are giving empty set in this procedure. These numebrs are e.g.
Golden Ratio=(1+Sqrt[5])/2, Sqrt[2] what mean that if you need improove accuracy one decimal digit we have to uses new fraction with bigger integers as numerator and denominator. For all numbers sequence is that same also for 1/number.
%e A138373 a(1)=3 because 38/7 approximate Pentanacii constant to 2 and also to 3 digits
%t A138373 << NumberTheory`Recognize`
b = {}; a = {}; Do[k = Recognize[N[Sqrt[5]/2, n + 1], 1, x]; If[MemberQ[a, k], AppendTo[b, n], AppendTo[a, k]], {n, 1, 500}]; b (*Artur Jasinski*)
%Y A138373 A138335, A138336, A138337, A138338, A138339, A138343, A138366, A138367, A138368, A138369, A138370, A138371, A138372, A138374, A138375, A138376, A138377, A138378, A138378, A138379
%O A138373 1
%K A138373 ,base,nonn,
%A A138373 Artur Jasinski (grafix at csl.pl), Mar 17 2008