Add to n its second smallest non-divider. Loop
franktaw at netscape.net
franktaw at netscape.net
Thu Jun 26 09:37:06 CEST 2008
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,
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