A276656 Nearest integer to the inverse function of the factorial where the exponential of n is the argument.

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, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22, 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

Offset

1,1

Comments

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.

The asymptotic is not as good as an average order of the zeta zeros. That is given by the França-LeClair asymptotic in A273061.

The errors between the first few terms of the non-rounded version of this sequence and the Riemann zeta zeros are:

-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.

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.

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.

Links

Table of n, a(n) for n=1..90.

Mathematica

(*
In Mathematica 8, this is the essential command that describes the sequence:
a(n) = Round[InverseFunction[Factorial, 1, 1][Exp[n]]]
a(n) = Round[InverseFunction[LogGamma, 1, 1][n]] - 1
*)
nn = 90;
Monitor[a =
  Table[Round[InverseFunction[Factorial, 1, 1][Exp[n]]], {n, 1,
    nn}], n]
(* Uncomment the code below to see the phenomenon described in the comment section *)
(* Monitor[a=Table[N[InverseFunction[Factorial, 1, 1][Exp[n -1/2]]], {n, 1, nn}], n]
g1=ListLinePlot[a, PlotStyle->Red]
g2=ListPlot[b=Table[Im[ZetaZero[n]]/(2*Pi), {n, 1, nn}]]
Show[g1, g2]
ListLinePlot[b-a] *)

Crossrefs

Cf. A046654, A273061, A275341, A275579, A275737.

Sequence in context: A327705 A104135 A354535 * A373119 A309250 A365339

Adjacent sequences: A276653 A276654 A276655 * A276657 A276658 A276659

Keyword

nonn

Author

Mats Granvik, Sep 11 2016

Status

approved