@**techreport**{RISC5321,author = {Jose Capco},

title = {{Odd Collatz Sequence and Binary Representations}},

language = {english},

abstract = {Here we investigate the odd numbers in Collatz sequences (sequences arising from the $3n+1$ problem). We are especially interested in methods in binary number representations of the numbers in the sequence. In the first section, we show some results for odd Collatz sequences using mostly binary arithmetics. We see how some results become more obvious in binary arithmetic than in usual method of computing the
Collatz sequence. In the second section of this paper we deal with some known results and show how we can use binary representation and OCS from the first section to prove some known results. We give a generalization of a result by Andaloro \cite{And2} and show a generalized sufficient condition for the Collatz conjecture to be true: If for a fixed natural number $n$ the Collatz conjecture holds for numbers congruent to $1$ modulo $2^n$ then the Collatz conjecture is true.},

year = {2015},

howpublished = {RISC Report },

length = {11},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}