  We develop a new closed-form arithmetic and recursive formula for the
partition function and a generalization of Andrews' smallest parts (spt)
function. Using the inclusion-exclusion principle, we additionally develop a
formula for the not-relatively prime partition function (which counts the
number of partitions that are not relatively prime). Moreover, we prove a
theorem involving the greatest common divisor of partitions, which allows us to
link partitions to prime numbers and lets us derive a formula for the
relatively prime function. Lastly, we develop numerous new identities for
Jordan's totient function of second order, Euler's totient function, and
Dedekind's psi function.
