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A permutation of the positive integers (but please note the starting offset: 0-indexed).

This sequence is a variant of A052330.

Shares with A064736, A302350, etc. the property that a(n) is either a divisor or a multiple of a(n+1). - Peter Munn, Apr 11 2018 on SeqFan-list. Note: A302781 is another such "divisor-or-multiple permutation" satisfying the same property. - Antti Karttunen, Apr 14 2018

The offset is 0 since S_0 = {1} denotes the first 2^0 = 1 terms. - Daniel Forgues, Apr 13 2018

This is "Fermi-Dirac piano played with Gray code", as indicated by Peter Munn's Apr 11 2018 formula. Compare also to A303771 and A302783. - Antti Karttunen, May 16 2018

This sequence has connections with A113552 and A281978: each pair of consecutive terms contains a term that divides the other term.

The derived sequence A282304 gives some insights about the fractal nature of this sequence.

Conjectures:

- All prime numbers appear in this sequence, in increasing order,

- the derived sequence A282304 is unbounded,

- this sequence is a permutation of the natural numbers.

From Antti Karttunen, May 17 2018: (Start)

The greedy algorithm which constructs this sequence can be understood also in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition {s1+...+sk} via mapping a(n) = prime(s1)*...*prime(sk), where s1 .. sk are the summands of an integer partition. The choices for constructing the next partition are: either remove some parts from the partition, but with the constraint that if any summand k is removed, then all copies of k present in partition must be removed in too. One may remove all copies of several distinct summands as well. If by such a removal of parts we can find any smaller partitions that have not yet occurred in the sequence, then we choose the one which has the smallest Heinz encoding value to be a(n+1). On the other hand, if all partitions obtained by such removals have already occurred in the sequence, one must then add one or more parts to the current partition, but with the constraint that one is allowed to use only summands that do not already occur in partition (but any number of such summands may be used, also of more than one kind, as long as such summands are not already present in the partition that corresponds to a(n)). Of all such valid new partitions not already encountered, one with the smallest Heinz encoding value is chosen to be a(n+1). Compare this to the rules given for similar A304531 and A303751.

Primes 2 .. 61 occur at: 2, 4, 8, 14, 34, 96, 193, 386, 770, 1538, 3074, 14647, 30533, 60824, 122349, 245225, 688293, 1535694.

Terms just before primes are: 1, 6, 20, 420, 1848, 6552, 556920, 1511640, 6953544, 11090902680, 26447537160, 444799488600, 411767273946600, 1361999444592600, 448097817270965400, 2159016755941924200, 768250528363503385200, 3827047701385526108400.

Primorials (A002110) occur at: 1, 2, 3, 10, 23, 56, 151, 343, 728, 1497, 3034, 6107, 20753, 51285, 112674, 235085, 655721, 1525973, 3151033, ...

Powers of 2: 2 .. 32 occur at: 2, 6, 26, 6531, 1210614, and immediately following terms are: 6, 20, 24, 48, 96.

Immediately preceding terms are: 1, 12, 840, 1163962800, 1479723952477818247200. After 1 these factor as: (2^2 * 3^1), (2^3 * 3^1 * 5^1 * 7^1), (2^4 * 3^2 * 5^2 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1), (2^5 * 3^2 * 5^2 * 7^2 * 11^1 * 13^1 * 17^1 * 19^1 * 23^1 * 29^1 * 31^1 * 41^1 * 43^1 * 47^1 * 53^1).

Observed recurrences: From n>=4 and k>=2 onward, there is a following general pattern:

For n = x .. x+(y-1), a(n) = prime(1+k)*a(n-(x-1)),

where y is the k-th record in A282304, and x is the position of that record in A282304, starting from the k = 2nd record in that sequence:

For n = 8 .. 8+4, a(n) = 5*a(n-7).

For n = 14 .. 14+10, a(n) = 7*a(n-13).

For n = 34 .. 34+30, a(n) = 11*a(n-33).

For n = 96 .. 96+89, a(n) = 13*a(n-95).

For n = 193 .. 193+184, a(n) = 17*a(n-192).

For n = 386 .. 386+382, a(n) = 19*a(n-385).

For n = 770 .. 770+766, a(n) = 23*a(n-769).

For n = 1538 .. 1538+1534, a(n) = 29*a(n-1537).

For n = 3074 .. 3074+3070, a(n) = 31*a(n-3073).

For n = 14647 .. 14647+11104, a(n) = 37*a(n-14646).

For n = 30533 .. 30533+29454, a(n) = 41*a(n-30532).

For n = 60824 .. 60824+30061, a(n) = 43*a(n-60823).

For n = 122349 .. 122349+91330, a(n) = 47*a(n-122348).

For n = 245225 .. 245225+121950, a(n) = 53*a(n-245224).

For n = 688293 .. 688293+367237, a(n) = 59*a(n-688292).

For n = 1535694 .. 1535694+596154, a(n) = 61*a(n-1535693).

Note how this forces values like prime powers to gaps between. E.g. 49 = a(367278) occurs 103 steps after the subsection a(n) = 53*a(n-245224) has ended at 245225+121950 (= 367175), but before the next regular subsection a(n) = 59*a(n-688292) starts at 688293.

(End)

The greedy algorithm which constructs this sequence can be understood also in terms of Heinz encodings of partitions (see A215366): Any term a(n) corresponds to a particular integer partition {s1+...+sk} via mapping a(n) = prime(s1) * ... * prime(sk), where s1 .. sk are the summands of an integer partition. The choices for constructing the next partition are: If by removing any parts from the partition we can find any smaller partitions that have not already occurred in the sequence, then we choose the one which has the smallest Heinz encoding value. On the other hand, if all partitions obtained by such removals have already occurred in the sequence, then we add to the current partition the least number of copies of the least positive integer that is not yet a part of the partition (A257993), until a partition is found which is not yet in the sequence.

From Antti Karttunen & David A. Corneth, May 01 - 04 2018: (Start)

No two successive descending terms, that is, a(n) > a(n+1) > a(n+2) never occurs.

For n > 1, if a(n) is odd then a(n-1) = 2^h * k * a(n) and a(n+1) = 2^j * a(n) for some h, k and j, that is, odd terms occur between two larger even numbers.

If a(n) < a(n+1) < a(n+2) then (a(n+1) / a(n)) is a divisor of a(n+2).

However, when a(n) < a(n+1) > a(n+2) then (a(n+1) / a(n)) might not be a divisor of a(n+2). The first such case occurs at n=64..66, as a(64) = 280 = 2^3 * 5 * 7, a(65) = 7560 = 2^3 * 3^3 * 5 * 7, and a(66) = 72 = 2^3 * 3^2. We have 7560/280 = 27, which is not a divisor of 72 (72/27 = 8/3).

In most cases, when a(n+1) < a(n) then gcd(a(n+1), a(n)/a(n+1)) = 1 (about 87% for the first 100000 descents). However, there are many exceptions to this, the first case occurring at a(65) = 7560 = 2^3 * 3^3 * 5 * 7 and a(66) = 72 = 2^3 * 3^2, with gcd(72,7560/72) = 3.

(End)

From David A. Corneth, May 04 2018: (Start)

The sequence can be partitioned into a tabf sequence with rows having the first element odd and the others even. It would give (1, 2, 6), (3, 12, 4, 36), (9, 18, 90), (5, 10, 30), (15, 60, 20, 180), (45, 360, 8, 24, 120, 40, 1080), (27, 54, 270), ...

It turns out that some rows are multiples of others; for example, the row (5, 10, 30) is five times the row (1, 2, 6). (End)

See also "observed scaling patterns" in the Formula section.

A303750 gives the positions of odd terms.

A282291 and A304531 are unitary divisor variants that satisfy the condition gcd(a(n+1), a(n)/a(n+1)) = 1, whenever a(n) > a(n+1).

The primes 2, 3, 5, 7, 11, 13, 19, 23 and 29 occur at positions 2, 4, 11, 42, 176, 1343, 8470, 57949, 302739, 1632898, thus after 7 and except for 13, a little earlier than they occur in variant A304531.

Consider A019565. Imagine that it is an automatic piano that "plays sequences" when an appropriate punched "tape" is fed to it (as its input), i.e., when it is composed from the right with an appropriate sequence p, as A019565(p(n)). The 1-bits in the binary expansion of each p(n) are the "holes" in the tape, and they determine which "tunes" are present on beat n. The "tunes" are actually primes that are multiplied together. Of course only "squarefree music" (sequences containing only squarefree numbers, A005117) is possible to generate this way, thus we call A019565 a "squarefree piano".

There is a more sophisticated instrument, called "Fermi-Dirac piano" (A052330), with which it is possible to produce sequences that may contain any numbers.

If the tape is constructed in such a way that between the successive beats (when moving from p(n) to p(n+1)), either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes occur simultaneously, then when fed as an input to either of these pianos, the resulting sequence "s" (the output) is guaranteed to satisfy the condition that s(n+1) is either a multiple or a divisor of s(n). For example, Gray code A003188 and its inverse A006068 are examples of such tapes, and they produce sequences A302033, A207901 and A284003, A302783.

This divisor-or-multiple permutation is obtained by playing "Fermi-Dirac piano" with the same tape which yields A303760 when "squarefree piano" is played with it. Note that A303760 is not a subsequence of this sequence, as its terms occur in different order than the squarefree terms here.

See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list and comments in A304537.

Presumably a(n) is a permutation of the positive integers.

Primes seem to occur in their natural order. 31 appears as a(7060). Primes p >= 37 are not found among the first 10000 terms.

Numbers n such that a(n)=n are 1, 2, 3, 12, 306, ...

A256918(n) = gcd(a(n), a(n+1)); gcd(a(A257120(n)), a(A257120(n)+1)) = gcd(a(A257475(n)), a(A257475(n)-1)) = n. - Reinhard Zumkeller, Apr 25 2015

For p prime: A257122(p)-1 = index of the smallest multiple of p: a(A257122(p)-1) mod p = 0 and a(m) mod p > 0 for m < A257122(p)-1. - Reinhard Zumkeller, Apr 26 2015

Each a(n) is always either a divisor or a multiple of a(n+1).

This sequence has similarities with A281978; here we look for substrings, there for divisors.

This sequence has similarities with A262323: in both sequences, consecutive terms overlap.