The "Shielded Primes" Sieve

[Giorgos Kalogeropoulos]

Hi everyone!

I would like to share a nice conjecture by  Angelos Ormpelidis  that I found at Rivera's primepuzzles pages.

It is about a sieve where you take the first k primes and then you remove sequentially pairs of primes with original gaps 1,2,4,6,8...

You can find the details here:  https://www.primepuzzles.net/conjectures/conj_111.htm

The conjecture is about the remaining primes and it states that if we take infinite primes the density is 1/e^2. 

I think this is a nice use of this constant that is not yet mentioned in  https://oeis.org/A092553 (should we add it?)

Also, should we add the "Shielded Primes" themselves as a new sequence?

 23, 37, 67, 113, 157, 173, 233, 263, 277, 317, 389, 409, 509, 577, 631, 719, 751, 887, 1039, 1109, 1129, 1223, 1237, 1283, 1297, 1307, 1327, 1423, 1447, 1511, 1543, 1559, 1613, 1693, 1709, 1733, 1847, 1889, 1913, 2003, 2099, 2137, 2179, 2273, 2333, 2357, 2399, 2423, 2447...

GK

[M F Hasler]

On Mon, May 4, 2026 at 11:50 AM Giorgos Kalogeropoulos <gap...@gmail.com> wrote:

a nice conjecture by  Angelos Ormpelidis  that I found at Rivera's primepuzzles pages.

about a sieve where you take the first k primes and then you remove sequentially pairs of primes with original gaps 1,2,4,6,8... :  https://www.primepuzzles.net/conjectures/conj_111.htm

The conjecture is about the remaining primes and it states that if we take infinite primes the density is 1/e^2. 

I think this is a nice use of this constant that is not yet mentioned in  https://oeis.org/A092553 (should we add it?)

I think before adding a remark to oeis.org/A092553 = 1/e² = 0.13533...

there should be more evidence for the conjecture, given that data up to 10^11 yields a value > 0.1355...

Also, Aitken's Delta² applied to the sequence given there,

0.133, 0.1373, 0.13493, 0.135529, 0.135801, 0.13568878, 0.1356289, 0.1355656, 0.1355249, ...

gives (0.13577..., 0.135408..., 0.136027..., 0.135721..., 0.135560..., 0.136730..., 0.135451...)

Although it looks slightly decreasing, we see oscillations between 0.1354 and 0.1357, with lower values not going below 0.1354 so far.

I don't mean to exclude the possibility, but IMHO we need more evidence to promote it from "guess" to "conjecture".

Also, should we add the "Shielded Primes" themselves as a new sequence? 

23, 37, 67, 113, 157, 173, 233, 263, 277, 317, 389, 409, 509, 577, 631, 719, 751, 887, 1039, 1109, 1129, 1223, 1237, 1283, 1297, 1307, 1327, 1423, 1447, 1511, 1543, 1559, 1613, 1693, 1709, 1733, 1847, 1889, 1913, 2003, 2099, 2137, 2179, 2273, 2333, 2357, 2399, 2423, 2447...

Yes, I think that would be a nice sequence.

-M.