Triangle numbers $T_k$ are integers of the form $\frac{k(k+1)} 2$.
A few triangle numbers happen to be perfect squares like $T_1=1$ and $T_8=36$, but more can be found when considering the product of two triangle numbers. For example, $T_2 \cdot T_{24}=3 \cdot 300=30^2$.

Let $S(n)$ be the sum of $c$ for all integers triples $(a, b, c)$ with $0<c \le n$, $c^2=T_a \cdot T_b$ and $0<a<b$. For example, $S(100)= \sqrt{T_1 T_8}+\sqrt{T_2 T_{24}}+\sqrt{T_1 T_{49}}+\sqrt{T_3 T_{48}}=6+30+35+84=155$.

You are given $S(10^5)=1479802$ and $S(10^9)=241614948794$.

Find $S(2^{N^2})$. Give your answer modulo $136101521$.