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Contrary to what might be expected (see comment after Proof), a(n) is not a permutation of all semiprimes; it is a permutation of even semiprimes {S_2} and semiprimes with smallest factor 3 {S_3}, plus {25, 35, 49, 77}. Proof (Start):

i. The sequence is infinite: we can always consider p*q for a(n+1), where p is the smallest factor in a(n-1) and q is the smallest prime > than the largest factor of any term already appearing in the sequence;

ii. a(15)=38 = 2*19 = 2*prime(8) and a(16)=33 = 3*11 = 3*prime(5);

iii. all {S_2} <= 38 and {S_3} <= 33 have appeared up to a(16), with 38 and 33 being maximum terms in {S_2} and {S_3}, respectively;

iv. all semiprimes with smallest factor >= 5 which are < 2*prime(9)=46 and 3*prime(6)=39 have appeared up to a(16). Consequently, the terms starting at a(17)=46 alternate between 2*prime(k) and 3*prime(k-3) k=9..infinity.

v. the only other numbers to have appeared are {25, 35, 49, 77}.

(End)

The above behavior is in contrast to A119718 (a permutation of all semiprimes because it lacks the constraint of a(n) being not coprime to a(n-2)). Interestingly, this sequence (A263648) shares the same essential rules as A098550 (the Yellowstone permutation) and many of its variations, while A119718 does not; one therefore might expect the opposite behavior to occur between this sequence and A119718. What observations or generalizations might we draw from this?

Proof that this is a permutation of the natural numbers follows the same basic format as the proof contained in A098550.

This sequence is one in a multitude of permutations of definable infinite sets (i.e., "infinite permutations") which share similar properties and similar proofs as A098550 (Yellowstone-type), and which are often (though not always - see for example A119718 and A255582) of the general form: a(n) is smallest number not yet appearing in the sequence which is coprime to a(n-1) and not coprime to a(n-2). But caution is warranted here: many sequences which may appear at first glance to be Yellowstone-type infinite permutations are not (e.g., A263648 is infinite, similar in structure to A119718 and even MORE similar to the general Yellowstone form, yet is not a permutation) or may not be provable in similar fashion (e.g., A254077, which is similar in structure to A255582 but cannot be demonstrated as infinite using Yellowstone-type constructions). What observations or generalizations might we draw from this?