  The Collatz Conjecture can be stated as: using the reduced Collatz function
$C(n) = (3n+1)/2^x$ where $2^x$ is the largest power of 2 that divides $3n+1$,
any odd integer $n$ will eventually reach 1 in $j$ iterations such that $C^j(n)
= 1$. In this paper we use reduced Collatz function and reverse reduced Collatz
function. We present odd numbers as sum of fractions, which we call `fractional
sum notation' and its generalized form `intermediate fractional sum notation',
which we use to present a formula to obtain numbers with greater Collatz
sequence lengths. We give a formula to obtain numbers with sequence length 2.
We show that if trajectory of $n$ is looping and there is an odd number $m$
such that $C^j(m) = 1$, $n$ must be in form $3^j\times2k + 1, k \in
\mathbb{N}_0$ where $C^j(n) = n$. We use Intermediate fractional sum notation
to show a simpler proof that there are no loops with length 2 other than
trivial cycle looping twice. We then work with reverse reduced Collatz
function, and present a modified version of it which enables us to determine
the result in modulo 6. We present a procedure to generate a Collatz graph
using reverse reduced Collatz functions.
