[VA] SRC#002- Almost integers and other beasties

[Valentin Albillo]

.
Hi, Gerson:

(12-14-2018 03:14 PM)Gerson W. Barbosa Wrote:  More or less in the same vein,

Thanks for your excellent findings, I was sure you'd never fail to contribute some amazing near-identities to this thread. As Bob Prosperi already pointed out, I too find this one particularly beautiful:

Quote:\[ \ln \left ( \frac{16\ln 878}{\ln \left ( 16\ln 878 \right )}\right )=3.14159265377 \]

Good finding indeed !

By the way, it's quite nice that the simple function x/Ln(x) sometimes gives almost-integer results for integer arguments (which means its graphic passes extremely close to integer-coordinates grid points), such as the following, in increasing order of "closeness":

             x               x/Ln(x)
          ---------------------------------
            17              6.0002541...
            163            31.9999987...
            53453        4910.0000012...
            110673       9529.0000006...
            715533      53078.0000004...

so that we have, for instance,

            53453/Ln(53453) = 4910.0000012...

In your case the argument x=16*Ln(878) results in x/Ln(x) being 23,1406926369... which is almost the famous Gelfond's constant = e^Pi (the easiest transcendental number to compute to high precision) so its natural logarithm is very nearly Pi itself.

Nice catch !

Have a fine weekend and best regards
V.
.