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Numbers m such that Sum_{k=1..m} (<1/2 + k*e> - <k*e>) > 0 where < > denotes fractional part.
4

%I #9 Feb 15 2021 21:23:20

%S 3,7,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24,25,27,28,29,31,35,39,

%T 41,42,43,45,46,47,48,49,50,51,52,53,54,55,56,57,59,60,61,63,67,81,85,

%U 87,88,89,91,92,93,95,99,113,117,119,120,121,123,124,125,127

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

%C See A194368.

%t r = E; c = 1/2;

%t x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]

%t y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]

%t t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];

%t Flatten[Position[t1, 1]] (* A194405 *)

%t t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 200}];

%t Flatten[Position[t2, 1]] (* A194406 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 200}];

%t Flatten[Position[t3, 1]] (* A194407 *)

%Y Cf. A194368.

%K nonn

%O 1,1

%A _Clark Kimberling_, Aug 24 2011