Product of sigma(k)/phi(k)

Paul D Hanna pauldhanna at juno.com
Fri Jul 8 20:21:10 CEST 2005

```I suggest that the CF for g.f.:

A(x) = 1 + Sum_{n>=0} 1/x^(2^n)

be submitted as the new sequence:

Name:
Value of constant terms of the partial quotients of the
continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n).

Sequence:
1,-1,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,-2,0,
2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,2,-2,0,2,0,0,
-2,0,2,-2,0,2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,
2,0,-2,2,0,0,-2,0,2,-2,0,2,0,-2,0,0,2,-2,0,2,0,0,-2,0,2,0,
-2,2,0,-2,0,0,2,-2,0,2,0,0,-2,0,2,-2,0,2,0,-2,0,0,2,0,-2,2,
0,0,-2,0,2,0,-2,2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,-2,0,2,0,-2,
0,0,2,0,-2,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,2,-2,0,2,0,0,-2,0,2,
-2,0,2,0,-2,0,0,2,-2,0,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,2,0,-2,
2,0,0,-2,0,2,-2,0,2,0,-2,0,0,2,-2,0,2,0,0,-2,0,2,0,-2,2,0,
-2,0,0,2,-2,0,2,0,0,-2,0,2,-2,0,2,0,-2,0,0,2,0,-2,2,0,0,-2,
0,2,0,-2,2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,-2,0,2,0,-2,0,0,2, ...

Example:
[1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, ...].

I do not find this in the OEIS.

Paul

-- "Paul D Hanna" <pauldhanna at juno.com> wrote:

To be specific, the continued fraction is:

[1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, x + 2, x,
x - 2, x, x, x + 2, x, x - 2, x + 2, x, x, x - 2, x, x + 2,
x - 2, x, x + 2, x, x - 2, x, x, x + 2, x, x - 2, x + 2, x,
x, x - 2, x, x + 2, x, x - 2, x + 2, x, x - 2, x, x, x + 2,
x - 2, x, x + 2, x, x, x - 2, x, x + 2, x - 2, x, x + 2, x, ...]

Is this already in the OEIS?

-- Ralf Stephan <ralf at ark.in-berlin.de> wrote:
> (Does sum {1/x^(2^i)} have a known formula?)

It has continued fraction coefficients satisfying a
bifurcating (d&c) recurrence, for integer x. I conjecture it
is possible to write the cont.frac. in the unknown using
such a recurrence, if Shallit and/or van der Poorten have