cp's OEIS Frontend

The first differences are 4, 6, 6, 6, 8, ... and apart from the initial term4, appears to coincide with A155067, the differences between successive odd-indexed primes. If confirmed, this will be one of the few formulas known for A307720.

The other bisection of A348773, A348775, seems much more mysterious.

It follows from the definition of A307720 that if p = k-th prime, k>1 and k odd, and q = (k+1)st prime, then the first time p appears in the sequence is at the start of a subsequence (p,1) [(p-1)/2 times], p, (1,q) [(q+1)/2 times].

For example, the fifth prime (11) first appears in A307720 at step 1374 at the start of the subsequence 11, 1, 11, 1, 11, 1, 11, 1, 11, 1, 11, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13, 1, 13.

So q appears p+1 steps after p, which explains why the terms of the present sequence appear in pairs.

In fact, it appears that one can make a much stronger statement about what happens immediately after the first occurrence of p. Look at the terms in A307720 following the first 11 at step 1374. It may be that the next O(p^2) terms are forced.