cp's OEIS Frontend

In other words, if the squarefree kernel of a(n-1) is a primorial term then a(n) is the least novel multiple of the smallest prime which does not divide a(n-1). Otherwise a(n) is the least novel multiple of the product of all primes < gpd(a(n-1)) which do not divide a(n-1). Primes >= 5 arrive late (as least unused term), and a(k) is prime(m) iff a(k-1) is A002110(m-1). The pattern around a prime is P(k), prime(k+1), 2*P(k), m*prime(k+1) for some multiplier m, where P(k) = A002110(k). The sequence is conjectured to be a permutation of the positive integers, with primes in natural order.

A common mode in this sequence is alternation of squarefree semiprime q(j)*q(k), j < k, followed by P(k-1)/q(j). The alternation often occurs in runs such that each iteration increments k. Example: a(241..246): q(2)*q(17) -> P(16)/q(2) -> q(2)*q(18) -> P(17)/q(2) -> q(2)*q(19) -> P(18)/q(2). a(16539..16572) represents a run of 17 alternations. - Michael De Vlieger, Jul 17 2023

It follows from the definition that the sequence is infinite.

Let r(n) = a(n-2)*a(n-1).

If rad(r(n)) is a primorial, then every prime q < gpf(r(n)) divides r(n), so m = 1, the empty product, and a(n) = u, the smallest missing number in the sequence so far.

If rad(r(n)) is not a primorial, then m > 1, and significant spikes can occur in scatterplot when there are multiple primes < gpf(r(n)) which do not divide r(n) (e.g., a(12) = 105, a(15) = 385, a(21) = 15015).

The only way a prime can occur is as u.

The sequence is a permutation of the positive integers since no number appears more than once and m = 1 eventually introduces any number not already placed consequent to terms arising from m > 1.