Consider the power sum $$S_a(b)=1^{2b}+2^{2b}+\cdots+(3a-2)^{2b}.$$ Let $\nu_3(x)$ denote the $3$-adic valuation of $x$.

QUESTION 1. (milder) Is this true? $$\nu_3\left(\frac{S_a(1)}{S_a(b)}\right)=0.$$ QUESTION 2. Is this true? $\nu_3(S_a(b))=\nu_3(2a-1)$.

Consider the power sum $$S_a(b)=1^{2b}+2^{2b}+\cdots+(3a-2)^{2b}.$$ Let $\nu_3(x)$ denote the $3$-adic valuation of $x$.

QUESTION 1. (milder) Is this true? $$\nu_3\left(\frac{S_a(b)}{S_a(1)}\right)=0.$$ QUESTION 2. Is this true? $\nu_3(S_a(b))=\nu_3(2a-1)$.