Steven takes an arithmetic sequence with common difference d and a geometric sequence with common ratio r and adds them together, term by term, to form a new sequence (i.e., the first term of a new sequence is the sum of the first terms of the arithmetic and geometric sequences, etc). If the first three terms of the new sequence are, in order, 3, 8, and 15, and d and r are both known to be positive integers, what is the sum of the possible values of d?