RE Boss wrote: > http://www.jsoftware.com/jwiki/Essays/Collatz_Conjecture
Ah, thanks for this pointer. I remember the sequence being called "the wondrous numbers" in Godel Escher Bach, but I don't think I've ever encountered this name. I wrote: > I'm not satisfied with my solution to "cast out all factors of two". > A secondary nit to pick: I don't like the way I pick "n > 0 and odd" The Essay's verb kills two verbs with one stone, by treating "triple and increment" and "cast out a (single) power of two" as separate intermediate steps. That is, if N is even, it's halved (casting out a single power of two). Otherwise (if N is odd) it's tripled and incremented. Either way, the verb is then applied to its output (so if N were a power of 2, the triple-and-increment would never be invoked). Whereas my verb implements the sentence: Pick n > 0 and odd, multiply by three, add one, cast out all factors of two, cook 'til (d)one. atomically. If N is even, it's incremented, then it's tripled and incremented, then all powers of 2 are cast out (simultaneously), then verb is applied to its output (so even if N were a power of 2, the triple-and-increment would still be applied). So the two verbs, given the same input, can produce different output sequences. In general, the sequences produced by casting-out all powers simultaneously will be shorter than those produced by casting-out one power at a time. An obvious exception is when the input is a power of 2 (see above). I believe the difference in formulation is due to the different English descriptions of the problem from which the verbs were transliterated. I'm still interested in verbs which can cast-out all factors of a number (narrowly: a prime number; broadly: any integer) simultaneously. -Dan ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
