[seqfan] Davide Rotondo's conjecture

[Neil Sloane]

Dear Sequence Fans,

On March 23 2024, Davide Rotondo sent me an email with the following
interesting conjecture.

(I've simplified it a bit.)

For a positive integer n, define a sequence b by b(0) = n; b(i) = n -
pi(b(i-1)) for i >= 1,

where pi(x) = number of primes <= x.

The conjecture is that after some initial terms, b becomes periodic with
period length 1 or 2,

 and the n for which the period is 2 are 3 together with A095116, that is,
2, 3, 4, 7, 10, 15, 18, 23, 26, 31, 38, 41, 48, ...

I've checked a few thousand terms and it seems to be true. Here is a random
example: for n = 2954, the b sequence is 2954, 2529, 2585, 2578, 2579,
2578, 2579, ..., with period 2.

There is probably a simple proof, but I don't have time to work on it.

I'm going to add it as a comment to A095116, and if anyone has a proof, I
will add it there.


Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com