I noticed the sequence A003064: 2,3,5,7,11,19,29,47,71,127,191,379,607,1087,1903,3583,6271, 11231,18287,34303,65131,110591,196591,357887 Name: Smallest number with addition chain of length n. 1903 = 11*173 is the first composite, but after that it's about 50:50. It looks like the limiting ratio might gradually approach 2. I get 2 3 5 7 11 19 29 53 87? for the corresponding addition-subtraction chain. (I'm working by hand, errors all too likely.) Rich rcs@cs.arizona.edu