Conjecture about the representation of a constant $C=0.6516...$

Question

It's a follow up of my previous question How to find the constant $C$ such that $f(x)\geq Cx$ :

We start with :

$$\left|\exp\left(1-\prod_{k=1}^{1000}\left(1-\frac{1}{2^{k}(k+1)}\right)\right)-\sqrt{2}-\frac{1}{2^{9}}-\frac{1}{2^{11}}-\frac{1}{2^{14}}-\frac{1}{2^{16}}+\frac{1}{2^{18}}-\frac{1}{2^{20}}-\frac{1}{2^{21}}-\frac{1}{2^{24}}+\frac{1}{2^{27}}+\frac{1}{2^{30}}-\frac{1}{2^{33}}+\frac{1}{2^{36}}+\frac{1}{2^{38}}+\frac{1}{2^{39}}-\frac{1}{2^{41}}\right|<4*10^{-13}$$

Conjecture

If $a_n\in N_{\geq 9}$:

$$\exp\left(1-\prod_{k=1}^{\infty}\left(1-\frac{1}{2^{k}(k+1)}\right)\right)=\sqrt{2}+\sum_{n=1}^{\infty}\pm\frac{1}{2^{a_n}}$$

I want to show that :

$$a_{n+1}>a_n$$

$$10>a_{n+1}-a_n$$

The $\pm$ means it's either positive or negative .

I have tried until $n=50$ without a counter-example .

As in my answer using the exponent I don't find it in oeis .

The $a_n$ seems also steady enought to believe in this conjecture .I have tried also to represent the power of two with an integral without a significant result .For example we have :

$$\int_{0}^{1}-\ln\left(x^{\frac{1}{2^{n+1}}}\left(1-x\right)^{\frac{1}{2^{n+1}}}\right)dx=1/2^n$$

We can also speak about the (ir)rationality of $C$.

Question :

How to (dis)prove it ?

Thanks in advance .

Answer 1

Strategy: Notice that $$ 2^{-2} = 2^{-1} - 2^{-2}, $$ using this fact you can always "fill the gaps" in the sequence $\{a_n\}$.

The following lemma is discussed in: Every real number can be represented as plus-minus consecutive terms of infinite geometric sequence $2^{-n}$.

Lemma. Every $C\in \mathbb R$ can be written as $ C = \sum_{n=k}^\infty \pm 2^{-n} $ for some $k\in \mathbb Z$.

The claim follows by applying the lemma to $$ C = \exp\left(1-\prod_{k=1}^{\infty}\left(1-\frac{1}{2^{k}(k+1)}\right)\right)-\sqrt{2}. $$