Totally to be expected. These are really the same thing. Let f(N) = (3N+1)/2^K, and g(N) = (3N-1)/2^K. Than a(N) = g(f(N)), b(N) = f(g(N)). So each family is the intermediate values from the other family. Actually, you can combine these into a single, signed function: h(N) = (1-3N)/2^K. Now, we have: 13 -> -19 -> 29 -> -43 -> 65 -> -97 -> 73 -> -109 -> 41 -> -61 -> 23 -> -17 -> 13. Franklin T. Adams-Watters -----Original Message----- From: cboyer@club-internet.fr After Dan's (a), I propose to define a second function (b): (a) M=(3N+1)/2^K then P=(3M-1)/2^K. (b) M=(3N-1)/2^K then P=(3N+1)/2^K. Interesting to see very similar properties: (a) two families, 1 and 17 (-> 19 -> 43 -> 97 -> 109 -> 61 -> 17...) (b) two families, 1 and 13 (-> 29 -> 65 -> 73 -> 41 -> 23 -> 13...) Similar percentages 1:17 for (a) and 1:13 for (b)??? Soon a report on the 2 functions from 1 to 10,000,000,000. Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.