Have you heard of reciprocal power sum?

Algebra Level 5

11+113+23+113+23+33+113+23+33+43+ \frac 1 1 + \frac 1 {1^3 + 2^3}+ \frac 1 {1^3 + 2^3 + 3^3 }+ \frac 1 {1^3 + 2^3 + 3^3 +4^3 } + \ldots

If the series above can be expressed as aπ2bc \frac {a \pi^2}{b} - c for positive integers a,b,ca,b,c with a,ba,b coprime. Find a+b+ca+b+c.

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