cp's OEIS Frontend

A permutation of the squarefree numbers A005117. The missing positive numbers are in A013929. - Alois P. Heinz, Sep 06 2014

From Antti Karttunen, Apr 18 & 19 2017: (Start)

Because a(n) toggles the parity of n there are neither fixed points nor any cycles of odd length.

Conjecture: there are no finite cycles of any length. My grounds for this conjecture: any finite cycle in this sequence, if such cycles exist at all, must have at least one member that occurs somewhere in A285319, the terms that seem already to be quite rare. Moreover, any such a number n should satisfy in addition to A019565(n) < n also that A048675^{k}(n) is squarefree, not just for k=0, 1 but for all k >= 0. As there is on average a probability of only 6/(Pi^2) = 0.6079... that any further term encountered on the trajectory of A048675 is squarefree, the total chance that all of them would be squarefree (which is required from the elements of A019565-cycles) is soon minuscule, especially as A048675 is not very tightly bounded (many trajectories seem to skyrocket, at least initially). I am also assuming that usually there is no significant correlation between the binary expansions of n and A048675(n) (apart from their least significant bits), or, for that matter, between their prime factorizations.

See also the slightly stronger conjecture in A285320, which implies that there would neither be any two-way infinite cycles.

If either of the conjectures is false (there are cycles), then certainly neither sequence A285332 nor its inverse A285331 can be a permutation of natural numbers. (End)

The conjecture made in A087207 (see also A288569) implies the two conjectures mentioned above. A further constraint for cycles is that in any A019565-trajectory which starts from a squarefree number (A005117), every other term is of the form 4k+2, while every other term is of the form 6k+3. - Antti Karttunen, Jun 18 2017

The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever x and y do not have a 1-bit in the same position, i.e., when A004198(x,y) = 0. See also A283475. - Antti Karttunen, Oct 31 2019

The above identity becomes unconditional if binary exclusive OR, A003987(.,.), is substituted for addition, and A059897(.,.), a multiplicative equivalent of A003987, is substituted for multiplication. This gives us a(A003987(x,y)) = A059897(a(x), a(y)). - Peter Munn, Nov 18 2019

Also the Heinz number of the binary indices of n, where the Heinz number of a sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and a number's binary indices (A048793) are the positions of 1's in its reversed binary expansion. - Gus Wiseman, Dec 28 2022

Essentially same as A091257 but computed starting from offset 0 instead of 1.

Each polynomial in GF(2)[X] is encoded as the number whose binary representation is given by the coefficients of the polynomial, e.g., 13 = 2^3 + 2^2 + 2^0 = 1101_2 encodes 1*X^3 + 1*X^2 + 0*X^1 + 1*X^0 = X^3 + X^2 + X^0. - Antti Karttunen and Peter Munn, Jan 22 2021

To listen to this sequence, I find instrument 99 (crystal) works well with the other parameters defaulted. - Peter Munn, Nov 01 2022

Equivalently, if binary expansion of n has m bits (say), compute derivative of n (A038554), getting sequence n' of length m-1; sort on n'.

Inverse of sequence A003188 considered as a permutation of the nonnegative integers, i.e., a(A003188(n)) = n = A003188(a(n)). - Howard A. Landman, Sep 25 2001

The sequence exhibits glide reflections that grow fractally. These show up well on the scatterplot, also audibly using the "listen" link. - Peter Munn, Aug 18 2019

Each bit at bit-index k (counted from the right hand end, with the least significant bit having bit-index 0) in the binary representation of a(n) is the parity of the number of 1's among the bits of the binary representation of n that have a bit-index of k or higher. - Frederik P.J. Vandecasteele, May 26 2025

Or, table of AND(i,j), i >= 0, j >= 0, read by antidiagonals. - N. J. A. Sloane, Feb 08 2016

Or, table of (i+j-Nimsum(i,j))/2 read by antidiagonals [Winning Ways, p. 75]. - N. J. A. Sloane, Feb 22 2019

This is a self-inverse permutation of the nonnegative integers.

The function "substitute x+1 for x" on polynomials over GF(2) is completely multiplicative.

What is the density of fixed points in this sequence? Do we get a different answer if we look only at irreducible polynomials?

From Antti Karttunen, Dec 27 2013: (Start)

As what comes to the above question, the number of fixed points in range [2^(n-1),(2^n)-1] of the sequence is given by A131575(n). In range [0,0] there is one fixed point: 0, in range [1,1] there is also one: 1, in range [2,3] there are no fixed points, in range [4,7] there are two fixed points: 6 and 7, and so on. (Cf. also the C-code given in A118666.)

Similarly, the number of cycles in such ranges begins as 1, 1, 1, 3, 4, 10, 16, 36, 64, 136, ... which is A051437 shifted two steps right (prepended with 1's): Because the sequence is a self-inverse permutation, the number of its cycles in range [2^(n-1),(2^n)-1] is computed as: cycles(n) = (A011782(n)-number_of_fixedpoints(n))/2 + number_of_fixedpoints(n), which matches with the identity: A051437(n-2) = (A011782(n)-A131575(n))/2 + A131575(n), for n>=2.

In OEIS terms, the above comment about multiplicativeness can be rephrased as: a(A048720(x,y)) = A048720(a(x),a(y)) for all integers x, y >= 0. Here A048720(x,y) gives the product of carryless binary multiplication of x and y.

The permutation conjugates between Gray code and its inverse: A003188(n) = a(A006068(a(n))) and A006068(n) = a(A003188(a(n))) [cf. the identity 1.19-9d: gB = Bg^{-1} given on page 53 of fxtbook].

Because of the multiplicativity, the subset of irreducible (and respectively: composite) polynomials over GF(2) is closed under this permutation. Cf. the following mappings: a(A014580(n)) = A234750(n) and a(A091242(n)) = A234745(n).

(End)

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

n XOR n-1, i.e., nim-sum of a pair of consecutive numbers.

Denominator of quotient sigma(2*n)/sigma(n). - Labos Elemer, Nov 04 2003

a(n) = the Towers of Hanoi disc moved at the n-th move, using standard moves with discs labeled (1, 3, 7, 15, ...) starting from top (smallest = 1). - Gary W. Adamson, Oct 26 2009

Equals row sums of triangle A168312. - Gary W. Adamson, Nov 22 2009

In the binary expansion of n, delete everything left of the rightmost 1 bit, and set all bits to the right of it. - Ralf Stephan, Aug 22 2013

Every finite sequence of positive integers summing to n may be termwise dominated by a subsequence of the first n values in this sequence [see Bannister et al., 2013]. - David Eppstein, Aug 31 2013

Sum of powers of 2 dividing n. - Omar E. Pol, Aug 18 2019

Given the binary expansion of (n-1) as {b[k-1], b[k-2], ..., b[2], b[1], b[0]}, then the binary expansion of a(n) is {bitand(b[k-1], b[k-2], ..., b[2], b[1], b[0]), bitand(b[k-2], ..., b[2], b[1], b[0]), ..., bitand(b[2], b[1], b[0]), bitand(b[1], b[0]), b[0], 1}. Recursively stated - 0th bit (L.S.B) of a(n), a(n)[0] = 1, a(n)[i] = bitand(a(n)[i-1], (n-1)[i-1]), where n[i] = i-th bit in the binary expansion of n. - Chinmaya Dash, Jun 27 2020

A self-inverse permutation of the natural numbers.

Nonnegative numbers rearranged with least disturbance to maintain a(n) not equal to n. - Amarnath Murthy, Sep 13 2002

Essentially lodumo_2 of A059841. - Philippe Deléham, Apr 26 2009

a(n) = A180176(n) for n >= 20. - Reinhard Zumkeller, Aug 15 2010

Take n, write it in binary, see what Rule 30 would do to that state, convert it to decimal: that is a(n). For example, we can see in A110240 that 7 = 111_2 becomes 25 = 11001_2 under Rule 30, which is shown here by a(7) = 25. - N. J. A. Sloane, Nov 25 2016

The sequence is injective: no value occurs more than once.

Fibbinary numbers (A003714) give all integers n>=0 for which a(n) = A048727(n) and for which a(n) = A269161(n).