A094065 Asymptotic form for prime.

0, 2, 5, 7, 10, 13, 16, 19, 22, 26, 29, 32, 36, 39, 42, 46, 49, 53, 57, 60, 64, 67, 71, 75, 78, 82, 86, 90, 93, 97, 101, 105, 109, 113, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 201, 205, 209, 213, 217, 221

Offset

1,2

Comments

This sequence results from a solution to a particular Laplacian of a linear perturbation associated with a Gaussian Dirichlet L-function used in a zeta zeros quantum Hamiltonian. The associated wave equation is: Psi(n, s) = (1+i)*exp(k_2 + k_1*s - s^2/(4*n)), where k_1 = (-4 + log(n))/4 and k_2 = n*log(n).

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

a(n) = floor(Re( n*(2 + log(n)/2 - sqrt((2*Pi + i*n)/(Pi*n))) )).

Mathematica

Table[Floor[Re[n*(2 +Log[n]/2 -Sqrt[I/Pi+2/n])]], {n, 1, 70}]

Prog

(PARI) {a(n) = floor( real(n*(2 + log(n)/2 - sqrt((2*Pi + I*n)/(Pi*n))) ))}; \\ G. C. Greubel, Mar 18 2019
(Magma) C<i> := ComplexField(); [Floor(Re( n*(2 + Log(n)/2 - Sqrt((2*Pi(C) + i*n)/(Pi(C)*n))) )): n in [1..70]]; // G. C. Greubel, Mar 18 2019
(Sage) [floor( (n*(2 + log(n)/2 - sqrt((2*pi + i*n)/(pi*n)))).real()) for n in (1..70)] # G. C. Greubel, Mar 18 2019

Crossrefs

Sequence in context: A067008 A376959 A189757 * A073593 A364451 A241510

Adjacent sequences: A094062 A094063 A094064 * A094066 A094067 A094068

Keyword

nonn

Author

Roger L. Bagula, May 31 2004

Extensions

Edited by G. C. Greubel, Mar 18 2019

Status

approved