"Blog" about A166133 by N. J. A. Sloane and others Feel free to add comments to this file (download it, add comments, reload it; it will then appear as a new file, but I can then replace the current version with the newer version). 1. Apr 01 2015. From me (NJAS) to Sequence Fans: A166133 is defined by After initial 1,2,4, a(n+1) is the smallest divisor of a(n)^2-1 that has not yet appeared in the sequence. There are a lot of huge outlying terms, but the bulk of the terms seem to lie roughly on a "straight" line. The sequence would die if all divisors of a(n)^2-1 were already present. There is a risk of this happening if a small number M (say) was missing for a long time, but eventually appeared, long after the average term had risen above M^2. So we might look at the smallest missing number after n terms, A256405, and ask how fast it grows (see also A256408, which simply lists the missing terms without repetitions). Reinhard Z. has produced a b-file of 10000 terms, but that is not enough to get a clear picture of the growth. It may be that A256405 also has linear growth, in which case there is less chance that A166133 dies. But if A256405 grows like sqrt(n), there could be trouble. Maybe someone could extend A256405 a very long way, so we get a better idea of its growth rate - enough so we can estimate how far we would have to go to see A246505(n) around sqrt(A166133(n)), if that ever happens. Alternatively, of course, find a proof that A166133 is infinite! ------------------------------------------ 2. Maximilian Hasler (who contributed the putative inverse permutation A255833), Reinhard Zumkeller, and Ray Chandler replied. Reinhard observed that there is a very long run of 6158 in A256405 starting at 14000 terms. On Apr 03 2015, Ray said: The run of 6158 in A256405 turns out to have 3328 occurrences. I suggest A256408 could use a companion sequence of the position of the first occurrence of A256408(n) in A166133 so the growth could be seen with smaller b-files. This doesn't make it any easier to compute, just represented more compactly. [That companion sequence is now A256409.] ------------------------------------------ 3. Hans Havermann found a 2009 plot of 450000+ terms of A166133, which I added to the entry. I replied to him as follows: Wow! [I seem to have said that about other plots you've sent me] That upper band - really a straight line, I think - is amazing. It would be interesting to do some kind of "spectral analysis" of these terms. If we were to think about a straight (not log) plot, then most of the points - the heavy black band - will lie on one line, and then there is a lot of fuzziness, and then there is an upper line. What is the slope of the upper line? Maybe I'll ask one of my old statistician friends to comment. ------------------------------------------ 4. John Mason computed 2 million terms of A166133 and A256405. These files were too large to upload with the usual OEIS "edit" mechanism, but I managed to load them to the server by hand (with scp). They can now be seen in the entries for A166133 and A256405. John says he may be able to get ten million terms. ------------------------------------------ 5. Franklin T. Adams-Watters, Apr 03 2015: I would guess the top line is basically where a(n) is approximately n^2-1. There don't seem to be any cases where a(n) is significantly less than n, so a(n) = a(n-1)^2-1 will be approximately that line. This implies that any a(n) >> n will *not* have a(n+1) = a(n)^2-1, which seems likely. ------------------------------------------