cp's OEIS Frontend

For all i, j: a(i) = a(j) => A005811(i) = A005811(j).

Old name: Square array read by antidiagonals: T(i,j) = product prime(k)^(Ei(k) XOR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; XOR is the bitwise operation on binary representation of the exponents.

Analogous to multiplication, with XOR replacing +.

From Peter Munn, Apr 01 2019: (Start)

(1) Defines an abelian group whose underlying set is the positive integers. (2) Every element is self-inverse. (3) For all n and k, A(n,k) is a divisor of n*k. (4) The terms of A050376, sometimes called Fermi-Dirac primes, form a minimal set of generators. In ordered form, it is the lexicographically earliest such set.

The unique factorization of positive integers into products of distinct terms of the group's lexicographically earliest minimal set of generators seems to follow from (1) (2) and (3).

From (1) and (2), every row and every column of the table is a self-inverse permutation of the positive integers. Rows/columns numbered by nonmembers of A050376 are compositions of earlier rows/columns.

It is a subgroup of the equivalent group over the nonzero integers, which has -1 as an additional generator.

As generated by A050376, the subgroup of even length words is A000379. The complementary set of odd length words is A000028.

The subgroup generated by A000040 (the primes) is A005117 (the squarefree numbers).

(End)

Considered as a binary operation, the result is (the squarefree part of the product of its operands) times the square of (the operation's result when applied to the square roots of the square parts of its operands). - Peter Munn, Mar 21 2022

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

Note that the starting offset is 0, to align with A052330 and A207901.

Shares with A064736, A207901, A298480, A302350, A302783, A303771, etc. the property that a(n) is either a divisor or a multiple of a(n+1). Permutations satisfying such property are called "divisor-or-multiple permutations" in the OEIS, although Mazet & Saias call them "chain permutations" in their paper. [Edited by Antti Karttunen, Aug 26 2018]

One way to construct such permutations is by composing A052330 from the right with any such permutation like A003188 or A302846 where the binary expansions of a(n) and a(n+1) always differ by just a single bit-position.

Further permutations satisfying the same condition could be constructed from higher-dimensional versions (i.e., greater than 2) of Hilbert's space-filling curves, where the coordinates of each dimension would be Gray coded separately and then interleaved together. Permutation A207901 is essentially a one-dimensional variant of the same idea, while this is constructed from the 2-dimensional curve A163357, which is a Hamiltonian path on N X N grid.

See Peter Munn's A300012 for another idea for constructing such a permutation. - Antti Karttunen, Aug 26 2018

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.

A squarefree analog of A207901 (and the subsequence consisting of its squarefree terms): Each term is either a divisor or a multiple of the next one, and the terms differ by a single prime factor. Compare also to A284003.

For all n >= 0, max(a(n + 1), a(n)) / min(a(n + 1), a(n)) = A094290(n + 1) = prime(valuation(n + 1, 2) + 1) = A000040(A001511(n + 1)). [See Russ Cox's Dec 04 2010 comment in A007814.] - David A. Corneth & Antti Karttunen, Apr 16 2018