Add to n its second smallest non-divider. Loop

[franktaw at netscape.net]

This got me thinking about, instead, simply adding tau(n) to n.  It 
turns out that
this sequence is in the OEIS: it's A064491.

What I find interesting is the square members of this sequence.  These 
are
interesting because it is at these values that the sequence changes 
parity.
(For anyone not familiar with this result: the divisors of any n fall 
naturally into
pairs: d and n/d; only when n is a square there is one, the square 
root, that is
unpaired -- or paired with itself -- in this way.)

The squares in A064491, up to 100 million, are:

1,4,9,64,81,784,1521,29584,34225,132496,136161,4999696,5076009,
15492096,15547249

which are the squares of:

1,2,3,8,9,28,39,172,185,364,369,2236,2253,3936,3943

Neither of these is in the OEIS.

Probably these sequences (the squares and square roots) are infinite, 
but I don't
see any likelihood of proving it.  For the sequence here -- n -> 2n + 
tau(n) -- it is
likely that 1 is the only square in the sequence, which would make 1 
and 3 the
only odd numbers in the sequence.  Again, I don't see any approach 
likely to
actually prove this.

One more note: there is a conjecture in A064491 that the sequence is 
asymptotic
to c*n*log(n), with c=1.401....  (Conjecture by Benoit Cloitre, who is 
known for
making conjectures on insufficient evidence.)  I am almost certain that 
this
conjecture is not correct.  I would conjecture, on the contrary, that 
a(n) ~
n*log(n) (equivalently, there exists c_1 and c_2 s.t. c_1*n*log(n) < 
a(n) <
c_2*n*log(n) for sufficiently large n), but that a(n) is not asymptotic 
to c*n*log(n)
for any c.  In particular, the slope of (a(n+1) - a(n)) / log(n) is 
different in areas
where a(n) is even as compared with those where it is odd.

Franklin T. Adams-Watters

On Wed, Jun 25, 2008 at 17:57,  <franktaw at netscape.net> wrote:
This rule simply takes n to 2n + tau(n).

On Wed, Jun 25, 2008 at 12:02, Hans Havermann <pxp at rogers.com> wrote:
> < Add to n the n-th smallest number not dividing n >
  I think each term is twice the previous term plus the number of 
divisors of
 the previous term:
  {1, 3, 8, 20, 46, 96, 204, 420, 864, 1752, 3520, 7068, 14160, 28360, 
56736,
 113508, 227040, 454176, 908424, 1816944, 3633908, 7267828, 14535662,