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A194409 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=Pi and < > denotes fractional part.

%I A194409 #8 Feb 15 2021 20:01:22
%S A194409 6,8,12,16,18,24,88,94,96,100,104,106,112,114,118,122,124,130,208,214,
%T A194409 216,220,224,228,230,236,328,334,336,342,448
%N A194409 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=Pi and < > denotes fractional part.
%C A194409 Every term is even; see A194368.
%t A194409 r = Pi; c = 1/2;
%t A194409 x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
%t A194409 y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
%t A194409 t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
%t A194409 Flatten[Position[t1, 1]]         (* A194408 *)
%t A194409 t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 1500}];
%t A194409 Flatten[Position[t2, 1]]         (* A194409 *)
%t A194409 t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 1500}];
%t A194409 Flatten[Position[t3, 1]]         (* A194410 *)
%Y A194409 Cf. A194368.
%K A194409 nonn
%O A194409 1,1
%A A194409 _Clark Kimberling_, Aug 24 2011