%I #10 Jan 20 2023 22:49:17
%S 2,3,4,5,5,6,6,7,7,8,8,9,9,10,10,10,11,11,12,12,12,13,13,14,14,14,15,
%T 15,15,16,16,16,17,17,18,18,18,19,19,19,20,20,20,21,21,21,22,22,22,22,
%U 23,23,23,24,24,24,25,25,25,26,26,26,27,27,27,27,28,28,28,29,29,29,30,30,30,30,31,31,31,32,32,32,32,33,33,33,34,34,34,34
%N Nearest integer to the inverse function of the factorial where the exponential of n is the argument.
%C Conjecture: Limit_{n->oo} (Im(zetazero(n))/(2*Pi))/(the nearest integer to the inverse function of the factorial where the exponential of n is the argument) = 1. In other words, the non-rounded version of this sequence is asymptotic to imaginary part of the Riemann zeta zeros divided by 2*Pi.
%C The asymptotic is not as good as an average order of the zeta zeros. That is given by the França-LeClair asymptotic in A273061.
%C The errors between the first few terms of the non-rounded version of this sequence and the Riemann zeta zeros are:
%C -0.0628295, 0.182931, 0.0998766, 0.316127, 0.118029, 0.295418, 0.289761, 0.15857, 0.406467, 0.206441, 0.246699, 0.34284, 0.357547, 0.155911, 0.407421, 0.298123, 0.275056, 0.266823.
%C Only the first error is negative, the rest are positive. The error appears to increase, but the rate of change in the errors becomes smaller higher up in the sequence.
%C Changing the Mathematica command InverseFunction[Factorial, 1, 1][Exp[n]]] into InverseFunction[Factorial, 1, 1][Exp[n-1/2]]] gives a more constant error fluctuating above 1/2 for the first 90 terms of the sequence.
%t (*
%t In Mathematica 8, this is the essential command that describes the sequence:
%t a(n) = Round[InverseFunction[Factorial, 1, 1][Exp[n]]]
%t a(n) = Round[InverseFunction[LogGamma, 1, 1][n]] - 1
%t *)
%t nn = 90;
%t Monitor[a =
%t Table[Round[InverseFunction[Factorial, 1, 1][Exp[n]]], {n, 1,
%t nn}], n]
%t (* Uncomment the code below to see the phenomenon described in the comment section *)
%t (* Monitor[a=Table[N[InverseFunction[Factorial, 1, 1][Exp[n -1/2]]],{n,1,nn}],n]
%t g1=ListLinePlot[a,PlotStyle->Red]
%t g2=ListPlot[b=Table[Im[ZetaZero[n]]/(2*Pi),{n,1,nn}]]
%t Show[g1,g2]
%t ListLinePlot[b-a] *)
%Y Cf. A046654, A273061, A275341, A275579, A275737.
%K nonn
%O 1,1
%A _Mats Granvik_, Sep 11 2016