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The sequence differs from A352976 after twenty-six terms. See the examples below.

In the first 100000 terms the maximum run of even and odd terms is three and five respectively; it is unknown if these runs have a maximum number of terms or are unbounded. The fixed points beyond 2 are 15, 25, 35, and it is likely no more exist. The primes appear in their natural order, and it is conjectured that this is a permutation of the positive integers.

From Michael De Vlieger, May 08 2022: (Start)

Let u be the smallest missing number in a(1..n-1) and let record r = max(a(1..n-1)).

Theorem: For odd prime q, r = mq -> q. Proof: generally, q may either divide or be coprime to r, but since equality and coprimality are prohibited by definition, primes may only divide records. This implies q serves as local minima, hence u > 2 is always an odd prime and r > 1 is always even.

Since r is always even, r is nondecreasing and even numbers appear in natural order. Also, r = 2q -> q, similar to the Lagarias-Rains-Sloane chain in the EKG sequence but without the successor 3q.

Theorem: a(n) = k such that k is odd iff least prime factor q | k and q | r. Proof: sequence definition demands smallest missing number such that (r,k) > 1. Otherwise, a(n) = a(n-1) + 2, since (a(n-1), a(n-1) + 2) > 1, and all smaller even numbers have already appeared.

All even numbers and primes appear, and every odd composite k has a least prime factor that admits k into the sequence on the occasion of (r,k) > 1.

There are 3 trajectories in scatterplot for a(n) > 1. The trajectory with lowest apparent slope is that of the primes, i.e., local minima. The trajectory with highest slope is that of records, i.e., even numbers, and the remaining numbers are odd composites. (End)

The sequence contains no primes or prime powers other than the powers of 2. As the sequence starts with 2 and 4, these terms being the largest and second largest values, the following terms will be even. This pattern continues until a term equal to the product of two or more odd primes occurs that shares a factor with the previous two largest even values and is smaller than the largest value plus 2. It is not possible for this value to be a prime larger than 2, or a power of such a prime, as the two terms with which it must share a factor differ by 2. It therefore cannot be between them either so it must be less than the second largest even term. Thus the next term after this odd composite must still share a factor with the two largest even values, and this will be the largest value plus 2 or another smaller odd composite. Therefore two more even values eventually become the two largest terms again, and thus the pattern of the two largest even terms, differing by two, interrupted by odd composites continues. Therefore no primes or prime powers other than powers of 2 will occur.

In the first 200000 terms the maximum run of even and odd terms is twelve and seven respectively; it is unknown if these runs have a maximum number of terms or are unbounded. The fixed points beyond 2 in the same range are 573, 597, 633, 6487, 21865, 22115, although it is likely more exist.

The sequence contains long runs on even terms, differing by 2, and odd terms, differing by various small even numbers. These runs are often interrupted by a large prime that becomes the new largest term. As this and the previous largest term are typically much larger than any other value the sequence then begins a long series of steadily increasing values that share a factor with the sum of these two largest terms.

The sequence is conjectured to be a permutation of the positive integers, although it apparently takes many terms for some primes to appear, e.g., after 200000 terms 73 has not occurred. The primes do not occur in their natural order. Beyond the first three terms there are nine fixed points in the first 200000 terms, although it is likely more exist.

Although all primes likely appear they do not occur in their natural order, e.g., 17 appears before 13. In the range studied each time a prime appears, beyond the initial 2 and 3, the next term is a multiple of the same prime. The largest multiple in the first 500000 terms is eight, first occurring at a(446271) = 64403, a(446272) = 515224. It is unknown if this ratio is unbounded for large n. Similarly the smaller of the two terms before a prime is a multiple of the prime. The largest ratio found being seven, first occurring at a(446271) = 64403, the same term as above.

In the first 500000 terms there are thirty-eight fixed points - 1, 2, 3, 4, 14, 32, 85, ..., 3277, 8651, 9223. It is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.