cp's OEIS Frontend

The set of real numbers between 0 and 1 that contain no 1's in their ternary expansion is the well-known Cantor set with Hausdorff dimension log 2 / log 3.

Complement of A081606. - Reinhard Zumkeller, Mar 23 2003

Numbers k such that the k-th Apery number is congruent to 1 (mod 3) (cf. A005258). - Benoit Cloitre, Nov 30 2003

Numbers k such that the k-th central Delannoy number is congruent to 1 (mod 3) (cf. A001850). - Benoit Cloitre, Nov 30 2003

Numbers k such that there exists a permutation p_1, ..., p_k of 1, ..., k such that i + p_i is a power of 3 for every i. - Ray Chandler, Aug 03 2004

Subsequence of A125292. - Reinhard Zumkeller, Nov 26 2006

The first 2^n terms of the sequence could be obtained using the Cantor process for the segment [0,3^n-1]. E.g., for n=2 we have [0,{1},2,{3,4,5},6,{7},8]. The numbers outside of braces are the first 4 terms of the sequence. Therefore the terms of the sequence could be called "Cantor's numbers". - Vladimir Shevelev, Jun 13 2008

Mahler proved that positive a(n) is never a square. - Michel Marcus, Nov 12 2012

Define t: Z -> P(R) so that t(k) is the translated Cantor ternary set spanning [k, k+1], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3. - Peter Munn, Oct 30 2019

Positive numbers that can be written in balanced ternary without a 0 trit. - J. Hufford, Jun 30 2015

Let S be the set of terms. Define c: Z -> P(R) so that c(m) is the translated Cantor ternary set spanning [m-0.5, m+0.5], and let C be the union of c(m) for all m in S U {0} U -S. C is the closure of the translated Cantor ternary set spanning [-0.5, 0.5] under multiplication by 3. - Peter Munn, Jan 31 2022

See discussions at A190803, A191106. The sequence a=A191108 has closure properties: the positive integers in (2+A191108)/3 comprise A191108, as do those in (-2+A191108)/3.

From Peter Munn, May 13 2019: (Start)

The closure of {1} in the positive integers under reflection about 3^k, k >= 1.

Asymptotic density is 0.

Consider a Sierpinski arrowhead curve formed of edges numbered consecutively from 0 at its axis of symmetry. The m-th edge is contained in the boundary of the plane sector occupied by the arrowhead if and only if m or -m is in this sequence.

For k >= 0, a(2^k) = 2*3^k - 1 and {a(i)/(2*3^k) | 1 <= i <= 2^k} is the set of center points of surviving intervals at the k-th step of generating the Cantor set, and therefore the set of center points of deleted middle-third intervals at the (k+1)-th step.

Define t: Z -> P(R) so that t(n) is the translated Cantor ternary set spanning [(n-1)/2, (n+1)/2], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3.

(End)

Subsequences include A007051, A000244, A153773, A153774.

First generation: 1

2nd generation: 2, 3

3rd generation: 5, 6, 8, 9

4th generation: 14, 15, 17, 18, 23, 24, 26, 27

Does every generation contain a prime?

From Peter Munn, Feb 10 2022: (Start)

Consider a Sierpinski arrowhead curve formed of edges indexed consecutively from 0 at its axis of symmetry and aligned with an infinite Sierpinski gasket so that each edge is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. The numbers {4*a(n)-1 : n >= 1} (that is, 3, 7, 11, 19, 23, 31, 35, 55, 59, ...) index the edges that are contained in the boundaries of certain triangular regions: each is the first region encountered of each successively larger size that does not lie across the axis of symmetry.

Let S be the set of terms. Define c: N -> P(R) so that c(m) is the scaled and translated Cantor ternary set spanning [m-0.5, m], and let C be the union of c(m) for all m in S. C is the closure under multiplication by 3 of the scaled and translated Cantor ternary set spanning [0.5, 1.0].

(End)

Positive numbers whose balanced ternary expansions contain exactly one digit 1. - Rémy Sigrist, May 08 2022

The sequence extends to negative n by defining a(n) = a(-n).

For k >= 1 numbers 1..k occur with the same periodic and mirror symmetries as in ruler function A051064, in which k occurs 3 times more frequently than k+1. Here k occurs 3/2 times more frequently than k+1, precisely 2^(k-1) times in every 3^k terms. 0 has asymptotic density 0. Taking a trisection shows some scale symmetry, again comparable to ruler functions, as illustrated in the example section.

The links include a pin plot of a(0..162) aligned above an inverted plot of A051064 (the emphatic marking of 0's is significant). Between each n_k where A051064(n_k) = k >= 2 and the nearest n_k' where A051064(n_k') > k (or n_k' = 0 if nearer), there are 2^(k-2) indices where k occurs in this sequence, forming a 2^(k-2)-tuple. The 2^(k-2)-tuples have identical patterns and each has symmetry about an n_(k-1) where A051064(n_(k-1)) = k-1.

For a given k, the tuples described above are periodic with two per fundamental period, and the closest pairs of these tuples jointly form the pattern of one of the equivalent tuples for k+1. These patterns relate to the nonperiodic pattern for 0's and to the Cantor set as follows.

Let S_k be the sequence of positive indices at which k occurs, with 3^(k-2) subtracted when k >= 2. Given its ruler-type symmetries, S_k k >= 2 is determined by its first 2^(k-2) terms, which are the same as the first 2^(k-2) terms of S_i for i > k. The limiting sequence as k goes to infinity is S_0, which is A191108. {A191108(i)/(2*3^k) | 1 <= i <= 2^k} is the set of center points of the intervals deleted at step k+1 when generating the Cantor ternary set. This leads to the following scaling property.

Define c: Z -> P(R) so that c(n) is the scaled and translated Cantor ternary set spanning [n-1, n+1], and let C_k be the union of c(n) for all integer n with a(n) = k. Clearly C_1 consists of a scaled Cantor set repeated with period 3. (The set's two nonempty thirds occur at alternating intervals of 4/3 and 5/3.) For k >= 1, C_k is C_1 scaled by 3^(k-1), consisting therefore of a scaled Cantor set repeated with period 3^k. C_0 is special: C_0 = (C_0)*3 = (C_0)/3 = -C_0. Specifically, (C_0)/2 is the closure of the Cantor ternary set under multiplication by 3 and by -1.

Take a Sierpinski arrowhead curve formed of unit edges numbered consecutively from 0 at its axis of symmetry and aligned with an infinite Sierpinski gasket so that each edge is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. If a(n) = 0, the n-th edge is contained in the sector boundary, otherwise the relevant triangular region seems to have side 2^(a(n)-1). See A307672 for a fuller description. The conjectured formulas below (that use A094373) derive from summing areas of regions within the gasket. - Corrected by Peter Munn, Aug 09 2019

From Charlie Neder, Jul 05 2019: (Start)

For each n, define the "2-balanced ternary expansion" E(n) as follows:

- E(n) begins with 0 or 1, according to the parity of n.

- The following digits are +, 0, or - as in standard balanced ternary, except + and - correspond to +2 and -2, respectively.

For example, we have E(4) = 0+-, E(7) = 10-, and E(13) = 1+-.

Then a(n) is the distance from the end of the rightmost 0, counting the last digit as 1, or 0 if 0 never appears. (End)

The length of row n is A173933(n). Observe that the m are actually less than k/3. Note that (k-m)/k is also in the Cantor set. If m appears in a row, then 3m does also. Let A and B be the first and last numbers in row n, then it appears that k = A + 3B. This implies A = k (mod 3). The interesting graph of this triangle shows that some ranges of m are not allowed.

When k is a prime of the form (3^r-1)/2, then the row consists of the 2^(r-1)-1 numbers (greater than 0) whose base-3 representation consists of only 0's and 1's. Hence, for r=3,7, and 13, the primes k are 13, 1093, and 797161, and the number of m < k/2 is 3, 63, and 4095.