cp's OEIS Frontend

The terms are concentrated along lines that contain numbers with a lowest prime factor of 2 or 3. Two of these lines are initially separated but join after approximately 130000 terms. This combined line then joins the uppermost line, which contains numbers with all prime factors and has a gradient of ~1.59, after approximately 680000 terms at which point a new series of smaller values appears. See the linked images.

Numbers with a larger number of prime divisors relative to the numbers close to it appear much later in the sequence. For example a(4014) = 16, a(14219) = 40, while even more delayed are a(685301) = 24 and a(704634) = 36. These last two appear after the above mentioned line merging after 680000 terms.

The lower lines containing terms with prime factors of 2 and 3 visible in the image of terms up to 1000000 are curving upward, possibly repeating the earlier behavior seen where similar lines eventually join with the uppermost line. If these do in fact eventually reach the uppermost line it is plausible this will once again signal the start of a new series of much lower valued terms.

The two lowest unseen numbers after 1000000 terms are 32 and 48, for former indicating that the longest run of consecutive odd values is only four after 1000000 terms. Although the sequence is conjectured to be a permutation of the positive integers, if these missing terms, especially those of the form 2^k, only appear after the recombination of the lower lines with the upper line then it may take an extraordinarily large number of terms for some values, like 2^k for large k, to eventually appear.

In the first 1000000 terms the only fixed points are the first three terms along with 14 and 15. It is possible more exist if the above mentioned upward trend of smaller values does occur.

The terms are concentrated along various lines that contain numbers with a lowest prime factor of 2, 3 or 5. These lines appear to have a slight upward curvature. However the uppermost line, which has a gradient of ~1.22, contains numbers with all prime factors. See the linked images.

Numbers with a large number of divisors relative to the numbers close to it appear much later in the sequence. For example a(96) = 6, a(1873) = 12, a(2328) = 18, a(192) = 16. The sequence is conjectured to be a permutation of the positive integers although it may take a very large number of terms for some values to appear, e.g., after 500000 terms numbers such as 24, 30, 36 have not occurred. In the same range the longest run of consecutive odd values is seven, while the only fixed points are the first three terms, although it is possible others exist for very large values of n if the smaller terms continue to increase relative to the uppermost line.

The sequence shows large jumps in value due to the sum occasionally forming a large prime, e.g., a(279) = 2650277753. The sequence is conjectured to be a permutation of the positive integers.

The sequence is conjectured to be a permutation of the positive integers. In the first 500000 terms the fixed points are 1, 2, 3, 4, 6, 9, 12. It is unlikely more exist although this is unknown.

The sequence is conjectured to be a permutation of the positive integers. In the first 150000 terms the fixed points are 1, 2, 3, 4, 6, 9, 12, 93, 6003, 6881, 16269, 100707, 114839, 116999. It is likely more exist.

The majority of terms are concentrated just below the line a(n) = n. However, some terms are much larger because the sum of the proper divisors of all previous terms is a prime number. In the first 10000 terms there are twenty-eight fixed points: 4, 5, 6, ..., 2486, 3280, 3292.

Conjecture: the sequence is a permutation of the positive integers.

The majority of terms are concentrated just below the line a(n) = n. However, some terms are much larger because the sum of the divisors of all previous terms is a prime number. In the first 5000 terms there are thirteen fixed points: 32, 52, 54, ..., 1331, 2082, 2097.

Conjecture: the sequence is a permutation of the positive integers.

Unlike A356850 all the terms are concentrated along three straight lines. In the first 100000 terms there are ten fixed points, 1, 2, 3, ..., 27, 57, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

The sequence is conjectured to be a permutation of the positive integers. In the first 10000 terms, other than the first three terms, there are thirty-five fixed points, the last being 2051. It is plausible no more exist although this is unknown.