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Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(13) and < > denotes fractional part.
4

%I #10 Feb 15 2021 02:18:29

%S 2,4,6,8,24,26,28,30,32,38,40,42,44,46,62,64,66,68,70,76,78,80,82,84,

%T 100,102,104,106,108,110,112,138,140,142,144,146,148,150,176,178,180,

%U 182,184,186,188,204,206,208,210,212,218,220,222,224,226,242,244

%N Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(13) and < > denotes fractional part.

%C See A194368.

%t r = Sqrt[13]; 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, 400}];

%t Flatten[Position[t1, 1]] (* A194392 *)

%t t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];

%t Flatten[Position[t2, 1]] (* A194393 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];

%t Flatten[Position[t3, 1]] (* A194394 *)

%Y Cf. A010470, A194368, A194392, A194394.

%K nonn

%O 1,1

%A _Clark Kimberling_, Aug 23 2011