Sequence A288640 in OEIS Riemann zeta zeros related

(*Definition:*)
FrancaLeClair[n_] = 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]];
Monitor[f =
    Table[Sign[Im[ZetaZero[n]] - FrancaLeClair[n]], {n, 1, 90}];, n];
(1 + f)/2
(*Comment 1:*)
Monitor[a =
    Table[Floor[
      Im[ZetaZero[n]]/(2*Pi)*Log[Im[ZetaZero[n]]/(2*Pi*Exp[1])] +
       11/8 - n + 1], {n, 1, 90}];, n];
a
(*Comment 2:*)
Monitor[Table[(1 -
     Sign[Im[Zeta[1/2 + I*2*Pi*E*Exp[LambertW[(n - 11/8)/E]]]]])/
   2, {n, 1, 90}], n]
(*Comment 3:*)
Table[Floor[
  2*(RiemannSiegelTheta[Im[ZetaZero[n]]]/Pi -
     Floor[RiemannSiegelTheta[Im[ZetaZero[n]]]/Pi])], {n, 1, 90}]
(*Comment 3:*)
(*For n>1:*)
Floor[2*FractionalPart[
   N[RiemannSiegelTheta[Im[ZetaZero[Range[90]]]]/Pi, 30]]]
(*Comment 4:*)
Clear[nn, n, k, t, FrancaLeclair, NumberOfZetaZeros];
nn = 89;
FrancaLeclair[n_] = 2*Pi*Exp[1]*Exp[ProductLog[(n - 11/8)/Exp[1]]];
NumberOfZetaZeros[t_] =
  RiemannSiegelTheta[t]/Pi + Im[Log[Zeta[1/2 + I*t]] + I*Pi]/Pi;
Monitor[b =
   N[Table[-(1 +
        2*Sum[(NumberOfZetaZeros[FrancaLeclair[k + 1]] -
             1) - (NumberOfZetaZeros[FrancaLeclair[k]] - 1) - 1, {k,
           1, n}]), {n, 0, nn}]], n];
(*Notice that sequence b is integer in itself despite the use of the \
Round function below*)
(1 + Round[b])/2
(*End*)