[seqfan] Franklin's A166133 - does it die?

[Neil Sloane]

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 on a straight line, roughly.

It 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).  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!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com