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A194405 Numbers m such that Sum_{k=1..m} (<1/2 + k*e> - ) < 0 where < > denotes fractional part.

%I A194405 #9 Feb 15 2021 22:32:52
%S A194405 1,5,33,37,65,69,71,72,73,75,76,77,79,83,97,101,103,104,105,107,108,
%T A194405 109,111,115,129,133,135,136,137,139,140,141,142,143,144,145,146,147,
%U A194405 148,149,150,151,153,154,155,157,161,165,167,168,169,171,172,173
%N A194405 Numbers m such that Sum_{k=1..m} (<1/2 + k*e> - <k*e>) < 0 where < > denotes fractional part.
%C A194405 See A194368.
%t A194405 r = E; c = 1/2;
%t A194405 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194405 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194405 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
%t A194405 Flatten[Position[t1, 1]]       (* A194405 *)
%t A194405 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];
%t A194405 Flatten[Position[t2, 1]]       (* A194406 *)
%t A194405 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];
%t A194405 Flatten[Position[t3, 1]]       (* A194407 *)
%Y A194405 Cf. A194368.
%K A194405 nonn
%O A194405 1,2
%A A194405 _Clark Kimberling_, Aug 24 2011