Comment on A354790
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This comment shall be an argument why no composites with more than three prime factors will be observed in A354790.
For a proof is it yet not rigorous enough, but hopefully this can be reached in future by formalizing it.


The sequence A354790 is closely related to A354169 as each prime factor may have only the state present or absent, analogous to the bits in A354169. The main difference is that in this sequence we will observe prime numbers more frequently, as we observe powers of two in A354169. This is a consequence of the Bertrand-Chebychev theorem, which states that for each m, a prime m < p < 2*m exists.
This means if we observe in A354790 a constellation such that if primes are interpreted as bits and the rules of A354169 are applied we would expect as the next term a number with two bits in A354169, we may observe in A354790 as the next terms one or more prime numbers instead before we observe a composite number.

Example regarding the difference between bits and prime factors:
2^0 < 2^1 < 2^0+2^1 < 2^2 < 2^0+2^2 < 2^1+2^2 ... 
but
2 < 3 < 5 < 2*3 < 7 < 2*5 < 11 < 13 ...

This has an interesting implication.
We know that in A354169 no number with Hamming weight greater than two will be observed.
We also know that if we were to stuff into some part of A354169 at random positions additional powers of two, but keep the powers of two in sequence and order and without duplicates, and then extend this modified sequence by the original algorithm of A354169, we would in no case observe numbers with Hamming weight greater than two, because we only increased the number of possible two-bit combinations which reuse a previously present single bit.
From this we can conclude that this sequence here will behave like A354169 in the sense that it may contain only composites with two prime factors.

More details for this:

In A354790 may only then a composite with more than three prime factors be observed, when any composites with three prime factors preceded this case, because otherwise a composite with three prime factors would have been an earlier choice.
This has the consequence that it is enough to consider composites with three prime factors here.

A composite C = P1*P2*P3 can only occur if two conditions are satisfied:
1.	P1*P2, P1*P3 and P2*P3 must appear already previously as part of the sequence.
2.	P1,P2,P3 must have their last usage already more than n/2 steps ago.

The same conditions hold for A354169 if we consider bits instead of primes.
In a proof by induction it was shown that this situation will never observed in A354169.
If we consider three bits B1, B2 and B3 in A354169, then it can be observed that whenever the situation is such that {B1,B2} and {B1,B3} where already previously seen in the sequence we will never observe {B2,B3} earlier than that any of the bits B1, B2 ,B3 was already reused in combination with a forth bit Bx which was not part of this initial set, this fact inhibits the possibility that both conditions 1 and 2 above can be fulfilled ad the same time.
If we compare A354790 with A354169 the situation for condition 1 is more restrictive as previously mentioned. As there are many primes which are a smaller choice over P1*P2*P3 we can say that not only P1*P2, P1*P3 and P2*P3 must appear already previously as part of the sequence but also at least some unknown count of further primes greater than P3. This will lead to the effect that there are further possible combinations like P1*P4 which will precede any possibility of P1*P2*P3.

The proof for A354169 provided by De Vlieger, Sigrist, Sloane, Trump et al. cannot be directly applied neither easily transformed to fit onto A354790, for two reasons. First the mentioned atoms require that the bits free for usage always reaches zero again, this is not the case in A354790, second the induction requires a predictable recurrence like pattern, but such regularity is here only limited observable as the irregularity of prime gaps introduces too much entropy.

So we need here a more general argument: 
From A354169 we did know that the bit patterns {B1, B2}; {B1, B3} and {B2, B3} could never be close enough such that {B1, B2, B3} could be possible, or in other words, before {B1, B2, B3} could happen any of these bits was already in combination with a fourth bit.
Our argument is now that in A354790 the density of such combinations can only be less so P1*P2, P1*P3 and P2*P3 cannot come closer in any case than {B1, B2}; {B1, B3} and {B2, B3} would be in the densest case of A354169, instead we get more single prime numbers interspersed into the sequence and such much more possibilities for more early combinations with new primes which are not part of the set.