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

%I #8 Feb 15 2021 20:01:03

%S 1,2,3,4,5,9,10,11,17,95,101,102,103,107,108,109,110,111,115,116,117,

%T 123,215,221,222,223,229,335

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

%C See A194368.

%t r = Pi; 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, 100}];

%t Flatten[Position[t1, 1]] (* A194408 *)

%t t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 1500}];

%t Flatten[Position[t2, 1]] (* A194409 *)

%t t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 1500}];

%t Flatten[Position[t3, 1]] (* A194410 *)

%Y Cf. A194368.

%K nonn

%O 1,2

%A _Clark Kimberling_, Aug 24 2011