A growth of triplets

Algebra Level 4

S=1123+1234+1345+1456+1567+S=\dfrac { 1 }{ 1\cdot 2\cdot 3 } +\dfrac { 1 }{ 2\cdot 3\cdot 4 } +\dfrac { 1 }{ 3\cdot 4\cdot 5 } +\dfrac { 1 }{ 4\cdot 5\cdot 6 } +\dfrac { 1 }{ 5\cdot 6\cdot 7 } +\cdots

If SS can be expressed in the form 1A\frac { 1 }{ A } with AA being a positive, composite integer, find AA.

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