Formula for finding the n-th prime number.

[Davide Rotondo]

Hi to all dear SeqFan members.

A little discovery about prime numbers (starting with the Buenos Aires constant).

Over the past few days, I've been playing with the constant:
C ≈ 2.9200509773161347...
the famous one that "contains" all the prime numbers.
And starting from my formula for (p_{n+1}), I managed to construct an explicit formula to directly obtain (p_{n+2})... and even a general version for (p_{n+k}).
---
Formula for (p_{n+2})
Let's define:
A = C Pₙ − pₙ ⌊C Pₙ₋₁ − 1⌋
(where (Pₙ = pₙ#), the primordial)
Then:
pₙ₊₂ = ⌊ ⌊A⌋ · (A − ⌊A⌋ + 1) ⌋
---
Fully expanded formula
pₙ₊₂ = ⌊ ⌊C Pₙ − pₙ⌊C Pₙ₋₁ − 1⌋⌋ ·
( C Pₙ − pₙ⌊C Pₙ₋₁ − 1⌋ − ⌊C Pₙ − pₙ⌊C Pₙ₋₁ − 1⌋⌋ + 1 ) ⌋
---
Generalization to (p_{n+k})
We define recursively:
A₁ = C Pₙ − pₙ⌊C Pₙ₋₁ − 1⌋
Aₖ₊₁ = ⌊Aₖ⌋ · (Aₖ − ⌊Aₖ⌋ + 1)
Then:
pₙ₊ₖ = ⌊Aₖ⌋

What do you think? Could I submit it to the encyclopedia?

Davide

[Simon Plouffe]

Hello, this constant appears here : https://arxiv.org/pdf/2010.15882

see also ; https://en.wikipedia.org/wiki/Formula_for_primes

they explain the recursive formula.

best regards, 
 Simon plouffe

--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/seqfan/28510a65-21b8-47e2-9c5c-8be09c17dc47n%40googlegroups.com.

[Charles Greathouse]

The constant is

https://oeis.org/A249270

in the OEIS. If you have anything to add, Davide, please do it there.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/d7b2838b-ad89-4810-a1c9-1913f9168562%40gmail.com.

[Davide Rotondo]

Thanks Simon and Charles.

Davide

You received this message because you are subscribed to a topic in the Google Groups "SeqFan" group.
To unsubscribe from this topic, visit https://groups.google.com/d/topic/seqfan/lfe7Irh99yY/unsubscribe.
To unsubscribe from this group and all its topics, send an email to seqfan+un...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/seqfan/CANXmBjxOKXqU4Yow8Xh%2BZu0jsRRt2UxPf1407OS2PdU59oZc%3Dw%40mail.gmail.com.