Numbers which have a "compact" form...

[Alonso Del Arte]

I don't have answers to your questions, but I want to suggest your
sequence be cross-referenced to A036057, the Friedman numbers, or
numbers that have a representation with operators and exactly the same
number of digits. At first one might assume that A036057 and this
sequence have no terms in common, but they do. Namely, 125, 128, 216,
343, 625, 1024, ...

Alonso

On Apr 1, 2005 12:42 PM, jb at brennen.net <jb at brennen.net> wrote:
> 
> Consider the sequence of positive integers which have a "compact"
> representation which uses fewer decimal digits than just writing
> the number out normally.  You are allowed to use the following
> symbols as well:
> 
>    ( )     grouping
>     +      addition
>     -      subtraction
>     *      multiplication
>     /      division
>     ^      exponentiation
> 
> (Not sure if it matters, but assume that you can use the
> unary minus sign, as well...)
> 
> If I'm not mistaken, the sequence would begin:
> 
>  125, 128, 216, 243, 256, 343, 512, 625, 729, 1000,
>  1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023,
>  1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032,
>  1033, ...
> 
>  (Note that 1015 to 1033 are all representable in the
>   form 4^5-d or 4^5+d, where d is a single digit.
>   1029 can also be written as 7^3*3.)
> 
> This sequence doesn't seem to be in the OEIS.
> 
> Does this sequence have density 1?  Or an even stronger
> conjecture:  Are there only a finite number of positive
> integers which are NOT in the sequence?  If so, can
> anything be determined about the largest integer without
> a "compact" representation?
> 
>  Jack
>