Formula for the $n$-th term of the sequence $1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$, where $f_n := \frac{1}{n} (f_{2n} - f_n)$ I'm struggling with this sequence.
$$1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$$
Where, $f_n := \frac{1}{n} (f_{2n} - f_n)$
You can also work it out for negative powers of 2,
$$f_\frac{1}{2} = \frac{2}{3}$$
$$f_\frac{1}{4} = \frac{8}{15}$$
$$f_{2^{-n}} = \prod_{k=1}^{n} (\frac{2^k}{2^k +1})$$
I wanted to know if it possible to find a formula for $f_n$, and to generalize it to all real numbers?
Can generating functions help here? (I don't know much about them.)
(And sorry if I had made some mistake)
 A: Using the q-Pochhammer symbol, if $f_1=1$,
$$ f_n=(-1;2)_{\frac{\log (n)}{\log (2)}}$$
If $n=2^k$, $f_k=(-1;2)_k$ which generates the sequence $A028361$ in $OEIS$
A: You asked about the sequence
$$1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$$
which is the OEIS sequence A171609.
It has a recursion with
$$f_1 = 1,\;\; f_{2n} = (n+1)f_n,\;\; f_{2n+1} = (n+1)f_{n+1}.$$
The question asked

I wanted to know if it possible to find a formula for $f_n$, and to generalize it to all real numbers?

I doubt very much there is a simple formula for $f_n$ depending on what
you consider a "formula". I also doubt very much that it can be extended
to all real numbers. However, as you indicated, it can be uniquely
extended to all positive
dyadic rationals. That is, rationals whose denominator is a power of $2$ using the
q-Pochhammer symbol as
$$ f_{k/2^n} := f_k/(-k/2;1/2)_n $$ if $k$ is any odd positive
integer and $n$ is any integer positive or negative.
In case you are interested, I wrote
PARI/GP
code to compute $f_n$ for dyadic rationals:
{f(n) = if(n<=0, 0, n==1, 1, frac(n), a(2*n)/(1+n), 
      n%2, (1+n)/2*a((1+n)/2), 1, (1+n/2)*a(n/2))};

