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.