Summing Specific Reciprocals

Algebra Level 3

\[1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{6}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{12}+\dfrac{1}{16}+\dfrac{1}{18}+\cdots = \, ?\]

Clarification: The denominators are all positive integers whose prime divisors are either 2 and/or 3 only. In other words, they exclude all numbers with a prime factor greater than 3.

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