Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors

Theorem
The arithmetic sequence:
 * $30, 66, 102, 138$

is the smallest of $4$ terms which consists entirely of positive integers each with $3$ distinct prime factors.

Proof
We demonstrate that this is indeed an arithmetic sequence:

demonstrating the common difference of $36$.

Then we note:

It is understood that smallest arithmetic sequence means:
 * integer sequence of length $4$ whose terms are in arithmetic sequence such that their sum is the smallest of all such sequences.

To demonstrate that this sequence is indeed the smallest, according to this definition, we list all positive integers less than $138$, each with $3$ distinct prime factors.

They are:

One cannot pick any other $4$ numbers from the list to form an arithmetic sequence.

Hence any further such arithmetic sequence will contain larger terms whose sum will be consequently larger.