Playing with constants using binary representation of primes, relating to A051006:

[N A]

Hello all!

Recently I've come across a constant:
Sum{primes p} 2^-p. For the purposes of this post, I'll define this to be S. The sequence for this is A051006.

In binary, this is obviously just 1's placed at prime places. But what if we played with this constant?

If you square it, you end up with an interesting sum over how many ways there are to sum 2 primes:

(Sum{primes p} 2^-p)^2 = Sum{n>=4} r(n)2^-n

With r(n) being the number of ways to sum to n using 2 primes. Investigating this constant as a binary string would be similar to investigating the Goldbach Conjecture. I say similar because there's the issue of carrying if r(n) > 2.

What I'd like to propose to add to the OEIS are the following:

S^2, 1/(1-S), and the digit-wise AND operation of S and 1/(1-S).

I don't think there should be an issue with proposing S^2 as a decimal and/or binary sequence.

I'm not sure if 1/(1-S) and S & 1/(1-S) are as interesting. 1/(1-S) would count all the ways of summing all the primes, and the & is just a filter for such that match up with the original prime positions. Mildly interesting is that this new constant in binary also has blocks of 101's, twin primes, but that's neither here nor there.

What do you all think?

Thank you for your time,
Nour Abouyoussef

[Gareth McCaughan]

On 24/04/2026 05:00, N A wrote:
> Hello all!
>
> Recently I've come across a constant:
> Sum{primes p} 2^-p. For the purposes of this post, I'll define this to
> be S. The sequence for this is A051006.
>
> In binary, this is obviously just 1's placed at prime places. But what
> if we played with this constant?
>
> If you square it, you end up with an interesting sum over how many
> ways there are to sum 2 primes:

[etc.]

No comment on what should or shouldn't be in the OEIS, but a suggestion
to N A: If you find this idea interesting then you should look up the
_Riemann zeta function_ and _Dirichlet series_; there's a lot of very
beautiful (and in some cases alas also very difficult) mathematics in
this general area.

--
g