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@netscape.net> wrote: This rule simply takes n to 2n + tau(n). On Wed, Jun 25, 2008 at 12:02, Hans Havermann <pxp@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,