A series of triplets

Calculus Level 5

\[S=\dfrac { 1 }{ 1\cdot 2\cdot 3 } +\dfrac { 1 }{ 4\cdot 5\cdot 6 } +\dfrac { 1 }{ 7\cdot 8\cdot 9 } +\dfrac { 1 }{ 10\cdot 11\cdot 12 } +\dfrac { 1 }{ 13\cdot 14\cdot 15 } + \cdots\]

If \(S\) can be expressed in the form \[\dfrac { \pi \sqrt { A } -A\ln { A } }{ 4A }\] with \(A\) being a positive, prime integer, find \(A\).


This problem is original.
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