  This paper highlights the emergence of the Omega sequence in number theory
and its connection with the emergence of a new prime number, and also
highlights its theoretical applications for Lucas-Lehmer primality test, and
Euclid-Euler theory for even perfect numbers. We also show that Omega sequences
unify and give new representations for Mersenne numbers, Fermat numbers, Lucas
numbers, Fibonacci numbers, Chebyshev sequence, Dickson sequence, and others.
