(This question is cross-posted on math.stackexchange)

I'm playing with p-adic valuations, and find that the odd harmonic sums, $\tilde{H}_k=\sum_{i=1}^{k}\frac{1}{2i-1}$, has 2-adic valuation $||k^2||_2=2||k||_2$.

E.g.) $\tilde{H}_4=\frac{176}{85}$ has $||\tilde{H}_4||_2=||176||_2=4=2||4||_2$.

$\tilde{H}_6=\frac{6508}{3465}$ has $||\tilde{H}_6||_2=2$. $||\tilde{H}_8||_2=||\frac{91072}{45045}||_2=6$. $\cdots$

How can I prove this?

(Some simple observation I have tried: As $||H_k||_2 =-r$ for $2^r\leq k <2^{r+1}$, $||H_k||_2=||H_{2k}||_2+1=||\frac{1}{2}H_k||_2$. Thus ultrametric ineq for $\tilde{H}_k=H_{2k}-\frac{1}{2}H_k$ doesn't help at all, yielding trivial result $\tilde{H}_k\in \mathbb{Z}_2$.)

(That $||\tilde{H}_k||_2=2||k||_2$ for odd $k$ is almost trivial, so the problem can be reduced to show $||\tilde{H}_{2k}||_2=||\tilde{H}_k||_2+2$, still have no idea)

(I've checked this holds up to $\sim 1000$ using Mathematica, and Mathematica spits the result almost immediately for $n\sim 1000$ and takes about a minute for $n\sim 10000$. )

For anyone interested : I've used the following Mathematica code, try it yourself

```
a = Table[
IntegerExponent[
Numerator[HarmonicNumber[2 n] - HarmonicNumber[n]/2], 2]/2, {n,
10000}];
b = Table [IntegerExponent[n, 2], {n, 10000}];
NonNegative[Min[a - b]]
```