Date: Fri, 11 Aug 1995 14:40:13 GMT
From: Erich Friedman 
Subject: writing numbers with 1's
Organization: Stetson University, DeLand, FL 32720
Newsgroups: rec.puzzles

since there seem to be a number of postings along the lines of:
"how do you make 113 with four 4's?"
i thought i would ask about a question that i am interested in.

consider writing numbers using only 1's.  allowed is + - * /
and concatination of 1's.  thus 37 can be written with six 1's.
(answer at end)

let m(n) be the minimum number of 1's needed to write n.
then m(n) is continuous in the sense that |m(n+1)-m(n)|<=1.
the behavior or m(n) is quite interesting.

let l(n) be the lowest number requiring n 1's.  how does l(n) grow?
in particular, what is lim_{n->oo} [ log l(n) ]/n ?