cp's OEIS Frontend

The sequence can only increase for two consecutive terms at most as if a(n) is even then a(n+1) will be a(n)/2, while if a(n) is odd and a(n+1) is prime then a(n+2) will be even and thus a(n+3) = a(n+2)/2.

For the first 100 million terms the lowest number not to have appeared is 888. It is likely all numbers eventually appear although this is unknown.

It is not known if this sequence cycles (that is, if it returns to 3).

This sequence is a variation of the Recamán sequence A005132 where the same rules apply except an additional restriction is added whereby a(n) = a(n-1) - n can occur only if gcd(a(n-1),n) > 1 or gcd(a(n-2),n) > 1, where gcd is the greatest common divisor. This additional restriction is inspired by the selection rules of A336957 and A098550.

Initially the sequence terms show a similar pattern to the Recamán sequence. However after about 1.5 million terms they begin to predominantly oscillate between two or a small number of values and the pattern of arching lines is no longer present. See the linked images.

It is unclear if all values are eventually visited; numerous small values like 4 and 5 have not occurred after 50 million terms.

The sequence excludes primes as otherwise the terms would simply be all the ordered integers >= 2. The terms appear to cluster around two lines; the lower line is a(n) ~ n while the upper lines starts with a gradient of approximately 2 and then slowly flattens. It is possible this gradient approaches 1 as n->infinity.

This is the digit sequence equivalent of the Enots Wolley sequence A336957. Like that sequence to avoid the sequence halting rapidly an additional rule is placed on a(n) - it must have as least one digit not in a(n-1). This implies a(n) cannot be a repdigit as otherwise a(n+1) would not exist. If this rule is removed then the sequence terminates after five terms: 1, 2, 20, 10, 11. The next term then does not exist as it must both contain and not contain the digit 1.

The sequence is probably infinite as any a(n) must contain at least two distinct digits, thus a(n+2) can have at most eight distinct digits. This implies that a(n+3) can always be created using a digit in a(n+2) and a digit not in a(n+2). However the behavior of the sequence as n gets very large is unknown.

The majority of terms are concentrated along a line whose slope is approximately 1.3. Occasionally though there are terms which correspond to the smallest unused number up to that point, and these tend to lead to a subsequent very large term. For example a(499) = 628, a(500) = 682, a(501) = 31, a(502) = 14322. Other large terms appear seemingly at random, for example a(15449) = 19880, a(15450) = 19099, a(15451) = 74962230.

It is likely all numbers > 1 eventually appear. The smallest number not seen after 20000 terms is 89.

Note that if the sequence starts with 2 then the terms are just all the increasing even numbers.

This sequence shows similar behavior to the EKG sequence A064413. See the linked image.

Similarly to A336957 no term a(n) can be a prime or a prime power as that would make it impossible to find a subsequent term a(n+1) that shared a prime factor with a(n) that a(n) did not share with a(n-1). Likewise each term a(n) must have at least one prime factor not in a(n-1) so that a(n+1) can share a prime factor with a(n) that is not in a(n-1).

At most two consecutive terms can be even, but the number of consecutive odd terms is likely unlimited, although this is unknown. Up to 500000 terms the longest run of consecutive odd terms is ten, the first occurring at a(106376).

For the terms studied it is found that each new prime that occurs in the prime factors that a(n) shares with a(n-1) appears in the natural order of all primes. It is likely this is true for all terms. Likewise for the terms studied the vast majority only share one prime factor with the previous term, although terms sharing multiple primes do occur. The first example is a(55) = 78 which shares prime factors 2 and 3 with a(54) = 90. The first terms to share three prime factors are a(8735) = 9630 and a(8734) = 9930, while the first to share four are a(248153) = 264810 and a(248152) = 265020.

Similar to A337181 the terms are concentrated along lines that have a slight downward curvature. See the first linked image. The lines are distinguished by containing terms with a different least prime factor (lpf) in a similar fashion to A336957. The top line contains those terms with lpf = 3, and then each lower line contains terms whose lpf is the next larger prime. These lines are interspersed with terms with lpf = 2, while the very lowest line contains the majority of terms with a large lpf. Interestingly the terms with lpf = 17 appear to form a line with a much higher concentration of terms than those with an lpf of 11, 13 or 19. See the second linked image.

The sequence is similar to A336957 but with the addition restrictions that each new term a(n) must share a 1-bit in its binary expansion with a(n-1), while sharing no 1-bits with the binary expansion of a(n-2). To ensure the sequence is infinite each a(n) must not only have a prime factor not in a(n-1), implying no prime or prime powers can occur (see A336957), it must also have a 1-bit in its binary expansion that is a 0-bit in the binary expansion of a(n-1).

A set-theory analog of A350359. This has the same relationship to A350359 as A115510 does to the EKG sequence A064413, as A252867 does to the Yellowstone permutation A098550, and as A338833 does to the Enots Wolley sequence A336957.

An equivalent definition in terms of sets: S(0) = {}, S(1) = {1}, S(2) = {1,2}; thereafter S(n) is the smallest set (different from the S(i) already defined) of positive integers such that S(n) meets S(n-2) but is disjoint from S(n-1) and S(n-3).