A153775 Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x are in S.

1, 2, 3, 5, 6, 8, 9, 14, 15, 17, 18, 23, 24, 26, 27, 41, 42, 44, 45, 50, 51, 53, 54, 68, 69, 71, 72, 77, 78, 80, 81, 122, 123, 125, 126, 131, 132, 134, 135, 149, 150, 152, 153, 158, 159, 161, 162, 203, 204, 206, 207, 212, 213, 215, 216, 230, 231, 233, 234, 239, 240, 242

Offset: 1

Author

Clark Kimberling, Jan 02 2009

Keywords

• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Peter Munn, Edges of Sierpiński arrowhead aligned with Sierpiński gasket.
• Eric Weisstein's World of Mathematics, Cantor Set
• Eric Weisstein's World of Mathematics, Closure

Crossrefs

Cf. A000244, A007051, A005836, A147992, A153773, A153774, A306556, A307744.

See also the related sequences listed in A191106.

Programs

Mathematica

nxt[n_] := Flatten[3 # + {-1, 0} & /@ n]; Union[Flatten[NestList[nxt,{1},5]]] (* G. C. Greubel, Aug 28 2016 *)

Formula

From Peter Munn, Feb 04 2022: (Start)

For k >= 0, 2^k <= n <= 2^(k+1)-1, a(n) = A005836(n+1) - (3^k-1)/2.

For n >= 1, A307744(4*a(2n)-1) = A307744(4*a(2n+1)-1) = A307744(4*a(n)-1) + 1.

(End)