cp's OEIS Frontend

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

Old name: a(n) = (1/3)*(s(n+1) - 1), where s = A026224.

Conjectures based on old name: these are numbers of the form (3*i+1)*3^j; see A182828, and they comprise the complement of A026179, except for the initial 1 in A026179.

From Peter Munn, Mar 17 2022: (Start)

Numbers with an even number of prime factors of the form 3k-1 counting repetitions.

Numbers whose squarefree part is congruent to 1 modulo 3 or 3 modulo 9.

The integers in an index 2 subgroup of the positive rationals under multiplication. As such the sequence is closed under multiplication and - where the result is an integer - under division; also for any positive integer k not in the sequence, the sequence's complement is generated by dividing by k the terms that are multiples of k.

Alternatively, the sequence can be viewed as an index 2 subgroup of the positive integers under the commutative binary operation A059897(.,.).

Viewed either way, the sequence corresponds to a subgroup of the quotient group derived in the corresponding way from A055047. (End)

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022

Is this A026140 shifted right? - R. J. Mathar, Jun 24 2025

Liouville function lambda(n) (A008836) is positive.

From Peter Munn, Jan 16 2020: (Start)

The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897, it forms a subgroup of the positive integers considered as a group under A059897.

This sequence is the intersection of A000379 and A056911, which are also subgroups of the positive integers under A059897.

(End)

The asymptotic density of this sequence is 2/Pi^2 (A185197). - Amiram Eldar, Oct 06 2020

Old name: 'Positions of zeros in A268387: numbers n such that when the exponents e_1 .. e_k in their prime factorization n = p_1^e_1 * ... * p_k^e_k are bitwise-xored together, the result is zero.

From Peter Munn, Sep 14 2019 and Dec 01 2019: (Start)

When trailing zeros are removed from the terms written in base p, for any prime p, every positive integer not divisible by p appears exactly once. This is the lexicographically earliest sequence with this property.

The closure of A238748 with respect to the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), the sequence thereby forms a subgroup, denoted H, of the positive integers under A059897(.,.). H is a subgroup of A000379.

(The symbol ^ can take on a meaning in relation to a group operation. However, in this comment ^ denotes the power operator for standard integer multiplication.) For any prime p, the subgroup {p^k : k >= 0} and H are each a (left and right) transversal of the other. For k >= 0 and primes p_1 and p_2, the cosets (p_1^k)H and (p_2^k)H are the same.

(End)

From Peter Munn, Dec 01 2021: (Start)

If we take the square root of the square terms we reproduce the sequence itself. The set of all products of a square term and a squarefree term is the sequence as a set.

The terms are the elements of the ideal generated by {6} in the ring defined in A329329. Similarly, the ideal generated by {8} gives A262675. 6 and 8 are images of each other under A225546(.), which is an automorphism of the ring. So this sequence and A262675, as sets, are images of each other under A225546(.). The elements of the ideal generated by {6,8} form the notable set A000379.

(End)

Dekking (2016) calls this the Toeplitz sequence or period-doubling sequence. - N. J. A. Sloane, Nov 08 2016

Fixed point of the morphism 1->12 and 2->11 (1 -> 12 -> 1211 -> 12111212 -> ...). - Benoit Cloitre, May 31 2004

a(n) is multiplicative. - Christian G. Bower, Jun 03 2005

a(n) is the least k such that A010060(n-1+k) = 1 - A010060(n-1); the sequence {a(n+1)-1} is the characteristic sequence for A079523. - Vladimir Shevelev, Jun 22 2009

The squarefree part of the even part of n. - Peter Munn, Dec 03 2020

The first column of the array is given by A123098; subsequent columns are obtained by applying the function A003961, i.e., replacing each prime factor by the next larger prime. - M. F. Hasler, Sep 17 2016

Interpretation with respect to A329329 from Peter Munn, Feb 08 2020: (Start)

With respect to the ring defined by A329329 and A059897, the first row gives powers of 3, the first column gives powers of 6, both in order of increasing exponent, and the body of the table gives their products. A329049 is the equivalent table in which the first column gives powers of 4.

A099884 is the equivalent table for the ring defined by A048720 and A003987. That ring is an image of the polynomial ring GF(2)[x] using a standard representation of the polynomials as integers. A329329 describes a comparable mapping to integers from the related polynomial ring GF(2)[x,y].

Using these mappings, the tables here and in A099884 are matching images: the first row represents powers of x, the first column represents powers of (x+1) and the body of the table gives their products.

Hugo van der Sanden's formula (see formula section) indicates that A019565 provides a mapping from A099884. In the wider terms described above, A019565 is an injective homomorphism between images of the 2 polynomial rings, and maps the image of each GF(2)[x] polynomial to the image of the equivalent GF(2)[x,y] polynomial.

(End)

These are the numbers from A005117 that are not divisible by 2 and 3.

Squarefree numbers coprime to 6. - Robert Israel, Sep 02 2016

Numbers k such that A008588(k) is in A005117. - Felix Fröhlich, Sep 02 2016

The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, May 22 2020

From Peter Munn, Nov 20 2020: (Start)

The products generated from each subset of A215848 (primes greater than 3).

Closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897. (End)

Multiplied by 6 we have 6, 30, 42, 66, 78, 102, ..., the values that may appear in A076978 after the 1, 2. [Don Reble, Dec 02 2020] - R. J. Mathar, Dec 15 2020

By the von Staudt-Clausen theorem, denominators of Bernoulli numbers are of the form 6*a(n) for some n. - Charles R Greathouse IV, May 16 2024

Numbers with an even number of even prime factors and an even number of odd prime factors.

The representation (as defined in A206284) of polynomials with nonnegative integer coefficients that are in the ideal of the polynomial ring Z[x] generated by x^2+x and 2.

The above property arises because the sequence lists the integers in the multiplicative subgroup of positive rational numbers generated by the squares of primes (A001248) and the products of two consecutive odd primes (A006094\{6}).

The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 4 and 15. For example, A003961(4) = 9, A003961(9) = 25, A003961(15) = 35, 15 * 35 = 525, 525/25 = 21. Alternatively, the sequence may be defined as the closure of A046337 under multiplication by 4.

From the properties of subgroups of the positive rationals we know that if we take an absent positive integer m and divide all terms that are multiples of m by m, we get all the integers in the same subgroup coset as m, and we can expect some of the nice properties here to carry over to the resulting set. Specifically, dividing the even terms by 2 gives all numbers with an odd number of prime factors that sum to an even number; dividing all terms divisible by an odd prime p by p, gives all numbers with an odd number of prime factors that sum to an odd number. The positive integers satisfying the 4th of the 4 possibilities are generated similarly, dividing by 6 (for example).

Numbers whose squarefree part is in A056913.

Term by term, the sequence is one half of its complement within A036349.

Self-inverse permutation of the natural numbers.