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

[Valentin Albillo]

(12-16-2018 09:06 AM)EdS2 Wrote:  Thanks, that leads to a rabbit hole of interesting links (OEIS and Mathoverflow.)

Interesting. I did not consult the OEIS for the results I gave above for x/Ln(x), I simply obtained them myself by running this trivial HP-71B program I wrote in J-F Garnier's Emu71 to quickly find them:

        1  DESTROY ALL @ M=1 @ I=2
        2  X=I/LN(I) @ N=ABS(X-IROUND(X)) @ IF N<M THEN M=N @ DISP I;,X
        3  I=I+1 @ GOTO 2

        >RUN

        2   2.88539008178
        5   3.10667467281
        9   4.09607651981
        13   5.06832618827
        17   6.00025410569
        163   31.9999987385
        53453   4910.00000122
        110673   9529.00000068
        715533   53078.0000004
        1432276   101044
        ...       ...

Substituting  X=I/LN(I) at line 2 by some other function and rerunning the program will result in a new set of almost-integer values, for instance:

        2 X=I/TAN(I) ...

        >RUN

         2   -.915315108721
         3   -21.0457576543
         7   8.03260795684
         37   -44.0072133321
         48   39.9957590124
         128   -123.004197859
         170   460.0010337
         1489   -12899.9995967
         2106   986.000155144
         11923   15493.9999873

i.e.: 1489/Tan(1489) = -12899.9995967 ~ -12900

and so on and so forth. Trivial variations of this trivial program will produce an infinitude of almost-integer valued expressions of all kinds.

Thanks for your interest and links.

V.
.