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A194410 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=Pi and < > denotes fractional part.

%I A194410 #8 Feb 15 2021 20:01:03
%S A194410 1,2,3,4,5,9,10,11,17,95,101,102,103,107,108,109,110,111,115,116,117,
%T A194410 123,215,221,222,223,229,335
%N A194410 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) > 0, where r=Pi and < > denotes fractional part.
%C A194410 See A194368.
%t A194410 r = Pi; c = 1/2;
%t A194410 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194410 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194410 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
%t A194410 Flatten[Position[t1, 1]]         (* A194408 *)
%t A194410 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 1500}];
%t A194410 Flatten[Position[t2, 1]]         (* A194409 *)
%t A194410 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 1500}];
%t A194410 Flatten[Position[t3, 1]]         (* A194410 *)
%Y A194410 Cf. A194368.
%K A194410 nonn
%O A194410 1,2
%A A194410 _Clark Kimberling_, Aug 24 2011