cp's OEIS Frontend

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001

This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.

Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005

4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005

Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011

The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019

The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

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

Like in binary Gray code A003188, also in this permutation the binary expansions of a(n) and a(n+1) differ always by just a single bit-position, that is, A000120(A003987(a(n),a(n+1))) = 1 for all n >= 0. Here A003987 computes bitwise-XOR of its two arguments.

When composed with A052330 this gives A302781.

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.

Like in binary Gray code A003188, also in this permutation the binary expansions of a(n) and a(n+1) differ always by just a single bit-position, that is, A000120(A003987(a(n),a(n+1))) = 1 for all n >= 0. Here A003987 computes bitwise-XOR of its two arguments. This is true for any composition P(A003188(n)), where P is a permutation that permutes the bit-positions of binary expansion of n in some way.

When composed with A052330 this gives a divisor-or-multiple permutation similar to A207901 and A302781.

Shares with A064736, A207901, A302781, A302350, etc. a property that a(n) is always either a divisor or a multiple of a(n+1). However, because multiple bits may change simultaneously when moving from A006068(n) to A006068(n+1) [with the restriction that the changing bits are all either toggled on or all toggled off], it means that also here the terms might be divided or multiplied by more than just a single Fermi-Dirac prime (A050376). E.g. a(3) = 3, while a(4) = A050376(1) * A050376(3) * 3 = 2*4*3 = 24. See also comments in A284003.

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.

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