number of $\lbrace 1,2,...,53 \rbrace$ subsets with member summation divisible by $3$ In $\lbrace 1,2,3,...,53 \rbrace$ how many subsections do we have with this condition:
the summation of subset members must be divisible by $3$.

for example $\lbrace 1,2 \rbrace$ & $\lbrace 1,2,3,4,5 \rbrace$ count.

The generating function for these is
$$\prod_{q=1}^{53} (1+z^q)$$
and with $\zeta = \exp(2\pi i/3)$ we obtain for the answer
$$\frac{1}{3} \sum_{p=0}^2 \prod_{q=1}^{53} (1+\zeta^{pq})
= \frac{1}{3} 2^{53}
+ \frac{1}{3} \prod_{q=1}^{53} (1+\zeta^{q})
+ \frac{1}{3} \prod_{q=1}^{53} (1+\zeta^{2q}).$$
Now since $(1+\zeta)(1+\zeta^2)(1+\zeta^3)= (2+\zeta+\zeta^2) \times 2
= 2$ this becomes
$$\frac{1}{3} 2^{53}
+ \frac{1}{3} 2^{17} (1+\zeta)(1+\zeta^2)
+ \frac{1}{3} 2^{17} (1+\zeta^2)(1+\zeta)
= \frac{1}{3} 2^{53} + \frac{2}{3} 2^{17}
\\ = 3002399751667712.$$
