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A194416 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) = 0, where r=sqrt(3) and < > denotes fractional part. 5

%I

%S 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,63,66,69,78,81,84,

%T 93,96,99,108,111,123,126,138,141,153,156,159,162,165,168,171,174,177,

%U 180,183,186,189,192,195,198,201,204,207,216,219,222,231,234

%N Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - <k*r>) = 0, where r=sqrt(3) and < > denotes fractional part.

%C Every term is divisible by 3; see A194368.

%t r = Sqrt[3]; c = 1/3;

%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, 150}];

%t Flatten[Position[t1, 1]] (* A194415 *)

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

%t Flatten[Position[t2, 1]] (* A194416 *)

%t %/3 (* A194417 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];

%t Flatten[Position[t3, 1]] (* A194418 *)

%Y Cf. A002194, A194368, A194415, A194417, A194418.

%K nonn

%O 1,1

%A _Clark Kimberling_, Aug 24 2011

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Last modified January 20 21:46 EST 2022. Contains 350472 sequences. (Running on oeis4.)