cp's OEIS Frontend

The sequence shows long runs of even terms differing by 2 which are eventually broken by a number with a product of odd primes less than the last even term. The term after such run-breaking terms is often significantly less than the previous terms, leading to the sequence showing abrupt dips in its values. In the first 200000 terms the longest even-numbered run is 106 terms, and it is likely these runs can grow arbitrarily long. Likewise long runs of odd terms also exist, the longest such run being 133 terms in the same range. However unlike the even-numbered runs which increase by 2 each term the odd-numbered runs increase with differing amounts between each term. Between the large dips in value the majority of terms are concentrated along a line with gradient ~ 1.125. See the linked images.

It takes many terms for the primes to appear, e.g. a(166) = 3, a(239) = 5, a(1841) = 23, a(13325) = 61, a(158205) = 191. They do not appear in their natural order.

Other than the first few terms the only fixed point up to 200000 terms is 63. It is possible more exist although this is unknown. The sequence is almost certainly a permutation of the positive integers.

As a(2) is even, which forces a(3) and a(4) to be odd, all following terms will be odd as the sum of two odd terms is even. Beyond a(5) = 7 all subsequent primes appear in their natural order.

The sequence shows large variations in its values due to some terms being required to equal the product of two large primes as all other divisors of both the sum and product have been used, e.g., a(25893) = 68485691. In the first 50000 terms there are 28233 occasions where the chosen term is a divisor of the sum and 22777 occasions where it is a divisor of the product. In the same range the fixed points are 1, 2, 3, 42, 3674, 11520, 18515, 39501 - it is likely more exist. The smallest number not to appear is 761, while the primes do not occur in their natural order. It is possible the sequence is finite as two terms could appear whose sum and product divisors have all been used. If not then the sequence is likely a permutation of the positive integers.

From Michael De Vlieger, Apr 03 2022: (Start)

Let S = a(n-1)+a(n+2) and let P = a(n-1)*a(n-2). Let u be the smallest missing number in a(1..n-1).

It is possible that S is prime, but P is prime iff either a(n-1)=1 or a(n-2)=1; since a(1) = 1 is given and followed by 2, for n > 2, P is always composite.

The axiom a(n) = (k | S) or (k | P) implies (k <= S) or (k <= P). Consequently, u <= k <= max(S,P). Let D contain divisors {d : d | S and d >= u} and let E contain factors {d : d | P and d >= u}. A solution k must appear in T = D U E.

For sufficiently large n, S is large, but P is large and composite.

In this sequence we may have equality of (one) input and output, since input S or P does not necessarily already exist in a.

(End)

In the first 500000 terms the smallest numbers that have not appeared are 15625, 25600, 28561, 36864. It is unknown if these and all other numbers eventually appear. In the same range on eighty-two occasions a(n) equals the sum of the previous two terms, these values begin 3, 5, 17, 64, 90, 73, 120, 144, 192.

See A355648 for the fixed points.

In the first 250000 terms the smallest numbers that have not appeared are 64, 1024, 11664, 15625. It is unknown if these and all other numbers eventually appear.

See A355637 for the fixed points.

The sequence produces numerous groupings of primes. For example a(3) to a(16) contains thirteen primes in fourteen terms, a(80) to a(102) contains fourteen primes in twenty-three terms. The sequences is conjectured to be a permutation of the positive integers.