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A194408 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=Pi and < > denotes fractional part.

%I A194408 #8 Feb 15 2021 21:23:09
%S A194408 7,13,14,15,19,20,21,22,23,25,26,27,28,29,30,31,32,33,34,35,36,37,38,
%T A194408 39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,
%U A194408 62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82
%N A194408 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) < 0, where r=Pi and < > denotes fractional part.
%C A194408 See A194368.
%t A194408 r = Pi; c = 1/2;
%t A194408 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194408 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194408 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
%t A194408 Flatten[Position[t1, 1]]         (* A194408 *)
%t A194408 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 1500}];
%t A194408 Flatten[Position[t2, 1]]         (* A194409 *)
%t A194408 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 1500}];
%t A194408 Flatten[Position[t3, 1]]         (* A194410 *)
%Y A194408 Cf. A194368.
%K A194408 nonn
%O A194408 1,1
%A A194408 _Clark Kimberling_, Aug 24 2011