cp's OEIS Frontend

Like A336957 no prime or prime power can be a term as if it shared a prime factor with the previous term it would then not contain a prime factor not in the previous term. It is likely all other composite numbers appear.

This sequence is a variant of A360519 where we only consider nonprime squarefree numbers (A000469).

Theorem: a(1) = 1, a(2) = 6; thereafter, a(n) is the smallest nonprime squarefree number m not yet in the sequence such that

(i) gcd(m, a(n-1)) > 1,

(ii) gcd(m, a(n-2)) = 1, and

(iii) m does not divide a(n-1).

Conjecture: The sequence is a permutation of A000469.

This is an analog of the Enots Wolley sequence A336957 based on binary representations rather than prime factorizations.

Let Ker(k), the kernel of k, denote the set of positions of 1's in the binary expansion of k. Thus Ker(15) = {0,1,2,3}, Ker(1) = {0}.

Theorem 1: For n > 2, a(n) is the smallest number m not yet in the sequence such that:

(i) Ker(m) intersect Ker(a(n-1)) is nonempty,

(ii) Ker(m) intersect Ker(a(n-2)) is empty, and

(iii) the set Ker(m) \ Ker(a(n-1)) is nonempty.

Say that a number k is a "candidate" for a(n) if properties (i), (ii) and (iii) hold, but k is not necessarily unused nor the lowest available number with those properties.

Define the "characteristic function" of a positive integer k by char_k(i) = 1 if Ker(a(i)) has a nonempty intersection with Ker(k), char_k(i) = 0 otherwise.

A property that this sequence shares with the Enots Wolley sequence is that when a new bit appears in the binary representation of a term for the first time, it must be as part of a number of the form 2^x + 2^y where 2^x < 2^y. In this situation, we say that 2^x is "introduced" by 2^y.

Theorem 2. If there are at least k distinct terms such that k is a candidate for a(i), k appears in the sequence.

Proof. If k is a candidate for a(i) but a(i) != k, either k has already appeared in the sequence and we have nothing to prove or there is some k' < k which is also a candidate. Since there are only k-1 positive integers less than k, this situation can occur at most k-1 times before k must be the lowest available candidate. QED.

Theorem 3. Every number with a binary weight of at least 2 appears in the sequence.

A proof is presented in the paper "The Binary Enots Wolley Sequence" by Nathan Nichols (see link).

This sequence uses the same rules as A064413 except with the additional restriction that a(n) must have a different number of divisors than a(n-1). Despite this additional restriction the overall behavior of the sequence is almost identical to A064413, with most values appearing only a few places away from their position in A064413.

The fixed points below 10000 are 1, 2, 8, 10, 14, 26, 32, 38, 259. It is likely no more exist.

After 200000 terms the smallest unused number is 28349, although like similar sequences this is almost certainly a permutation of the positive integers. In the same range the fixed points are 4, 21, 50, 1269; it is likely no more exist. See the linked image.

In other words, if a(n-1) has k prime divisors p_j, 1 <= j <= k which do not divide a(n-2), where 1 <= k <= omega(a(n-1)), and if m_j*p_j is the least multiple of p_j which is not already a term, then a(n) = Min{m_j*p_j; 1<=j<=k}. Otherwise every prime divisor of a(n-1) also divides a(n-2), in which case a(n) is the least multiple of the squarefree kernel of a(n-1) which is not already a term. If a(n-1) and a(n-2) are coprime the computation of a(n) ranges over all prime divisors of a(n-1). This happens only once (n=3), after which all adjacent terms share a common divisor (as in the EKG sequence, A064413).

Departs from A064413 and A352187 at a(19), a(31) respectively, and apparently shares the way odd primes are proven to appear in the former and conjectured to appear in the latter; namely as 2*p, p, 3*p.

Conjectures: Permutation of the positive integers with primes in natural order, appearing in same way as in EKG.

From Michael De Vlieger, Oct 22 2022 (Start)

An algorithm similar to the Rains algorithm for the EKG sequence efficiently generates the sequence.

Like the EKG sequence, this sequence forces primes into divisibility; Primes divide their predecessors and successors. Consequently they exhibit Lagarias-Rains-Sloane chain 2p -> p -> 3p outside of p = 2, just as in the EKG sequence.

Let us define several quasi-rays conspicuous in the scatterplot. From lowest to highest apparent slope, we have the following:

- beta: local minima, i.e., a(1)=1 and primes p in order.

- gamma: 2p, 4p, and certain other composites.

- alpha-k: k*p from large k to k = 3. This system appears as a series of fine quasi-rays, with 3p generally comprising records.

Records are 3p outside of {1, 2, 4, 12, 18, 24, 25, 30, 36, 42, 56, 70, 105}.

a(33) = 35 behaves like a prime; 70 precedes and 105 follows it. a(34) = 105 is conspicuous as it appears earlier than expected. (End)

The sequence uses a similar selection rule to the Yellowstone permutation A098550 but instead of choosing the smallest number that has not occurred earlier that has a common factor with a(n-2) and no common factor with a(n-1), the number closest to a(n-1) that satisfies these rules is selected for a(n). If two such numbers are the same distance from a(n-1) then the smaller is chosen.

As any number n is coprime to n-1 there are many such pairs of values differing by one in the sequence. If a(n-1) is even and a(n) is odd then the only time a(n+1) will not be a(n)-1 is if a(n)-1 has already appeared in the sequence.

The majority of terms are clustered along a line with gradient approximately 1.25 . However the line is broken into smaller regions, each region having a slightly higher gradient, by the terms dropping to much smaller values before returning to the main line. See the linked image.

Excluding the first five terms, in the first 5 million terms the maximum number of consecutive terms that increase is only two. This only occurs at five places, when n=9,25,79,13705,275345. In the same range there are many regions of consecutive decreasing terms, the longest being 5997 terms starting from n=1902153.

In the first 5 million terms the only fixed points, other than the first three terms, are 4 and 313362. As the terms for larger n seem to drop below the a(n)=n line on numerous occasions it is possible more exist, although this is unknown. The smallest number not appearing is 31. It is unknown is all values eventually appear. In the same range the largest change in consecutive terms is from a(3503960)=30982 to a(3503961)=3191249, a difference of 3160267.

The sequence preserves throughout the coprime relations found in the first four positive integers 1,2,3,4 (4 is prime to 1 and 3 but not to 2).

A prime term p at a(n) is necessarily preceded at a(n-2) by a multiple m*p of p, and followed at a(n+2) by a different multiple w*p of p (m,w > 1).

The sequence is infinite. Proof: For successive terms a,b,c,d we can choose a multiple e = q*c of c, where q is any prime which divides neither b nor d and such that e is not a prior term. Then e is prime to b and d but not to c, and since it has not been seen before we have at least one candidate for the term following d, which we choose as the least such number.

The definition implies that there can be no consecutive even terms (since then they would not be coprime). However, consecutive odd terms are not excluded, and do occur (eg 21 can follow 25 because they are coprime). Although two adjacent primes is possible, and does occur (a(9)=5, a(10)=7), three is not, since consecutive distinct primes p,q,r would imply gcd(p,r)>1.

Similar sequences with the same coprime relations as in 1,2,3,4 can be generated from any start terms a,b,c,d with b=a+1,c=b+1,d=c+1, provided a is congruent to 1 or 5 mod 6 (A007310).

Conjecture: The sequence is a permutation of the positive integers in which the primes appear in their natural order.

The sequence differs from A352976 after twenty-six terms. See the examples below.

In the first 100000 terms the maximum run of even and odd terms is three and five respectively; it is unknown if these runs have a maximum number of terms or are unbounded. The fixed points beyond 2 are 15, 25, 35, and it is likely no more exist. The primes appear in their natural order, and it is conjectured that this is a permutation of the positive integers.

From Michael De Vlieger, May 08 2022: (Start)

Let u be the smallest missing number in a(1..n-1) and let record r = max(a(1..n-1)).

Theorem: For odd prime q, r = mq -> q. Proof: generally, q may either divide or be coprime to r, but since equality and coprimality are prohibited by definition, primes may only divide records. This implies q serves as local minima, hence u > 2 is always an odd prime and r > 1 is always even.

Since r is always even, r is nondecreasing and even numbers appear in natural order. Also, r = 2q -> q, similar to the Lagarias-Rains-Sloane chain in the EKG sequence but without the successor 3q.

Theorem: a(n) = k such that k is odd iff least prime factor q | k and q | r. Proof: sequence definition demands smallest missing number such that (r,k) > 1. Otherwise, a(n) = a(n-1) + 2, since (a(n-1), a(n-1) + 2) > 1, and all smaller even numbers have already appeared.

All even numbers and primes appear, and every odd composite k has a least prime factor that admits k into the sequence on the occasion of (r,k) > 1.

There are 3 trajectories in scatterplot for a(n) > 1. The trajectory with lowest apparent slope is that of the primes, i.e., local minima. The trajectory with highest slope is that of records, i.e., even numbers, and the remaining numbers are odd composites. (End)

The sequence contains long runs on even terms, differing by 2, and odd terms, differing by various small even numbers. These runs are often interrupted by a large prime that becomes the new largest term. As this and the previous largest term are typically much larger than any other value the sequence then begins a long series of steadily increasing values that share a factor with the sum of these two largest terms.

The sequence is conjectured to be a permutation of the positive integers, although it apparently takes many terms for some primes to appear, e.g., after 200000 terms 73 has not occurred. The primes do not occur in their natural order. Beyond the first three terms there are nine fixed points in the first 200000 terms, although it is likely more exist.