A173934 -id:A173934 - cp's OEIS frontend

A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.

1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 61, 63, 73, 75, 79, 81, 163, 165, 169, 171, 181, 183, 187, 189, 217, 219, 223, 225, 235, 237, 241, 243, 487, 489, 493, 495, 505, 507, 511, 513, 541, 543, 547, 549, 559, 561, 565, 567, 649, 651, 655, 657, 667, 669, 673, 675, 703, 705, 709, 711, 721, 723, 727, 729, 1459, 1461, 1465, 1467, 1477

Offset: 1

Clark Kimberling, May 26 2011

Related sequences for various choices of i and k as defined in A190803:
  A003278:  (i,k) = (-2,-1)
  A191106:  (i,k) = (-2, 0)
  A191107:  (i,k) = (-2, 1)
  A191108:  (i,k) = (-2, 2)
  A153775:  (i,k) = (-1, 0)
  A147991:  (i,k) = (-1, 1)
  A191109:  (i,k) = (-1, 2)
  A005836:  (i,k) = ( 0, 1)
  A191110:  (i,k) = ( 0, 2)
  A132140:  (i,k) = ( 1, 2)
For a=A191106, we have closure properties: the integers in (2+a)/3 comprise a; the integers in a/3 comprise a.
For k >= 1, m = a(i), 1 <= i <= 2^k seems to be m such that m/(3^k+1) is in the Cantor set (except that m = 0 and m = 3^k+1 do not appear). For k >= 2,  m = (a(i)-1)/2, 1 <= i <= 2^k seems to be m such that m/((3^k-1)/2) is in the Cantor set. - Peter Munn, Jul 06 2019
Every even number is the sum of two (possibly equal) terms. More specifically: terms a(1) through a(2^n) = 3^n sum to even numbers 2 times 1 through 3^n. Every even number is infinitely often the difference of two terms. Since the sequence is equal to 2*A005836(n) + 1, these properties follow immediately from similar properties of A005836 for every number. - Aad Thoen, Feb 17 2022
if A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+2*3^n), similar to A003278. - Arie Bos, Jul 26 2022

			1 -> 3 -> 7,9 -> 19,21,25,27 -> ...

Ivan Neretin, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.
D. Jordan and R. Schayer Rational points on the Cantor middle thirds set, Penn State, REU 2003.
Eric Weisstein's World of Mathematics, Cantor Set

Cf. A005823, A005836, A054591, A088917 (characteristic function), A173934, A190803, A191108.
Partial sums of A061393.
Similar formula as A003278, A_(n+1)=(A_n,A_n+2*3^n).

h = 3; i = -2; j = 3; k = 0; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191106; regarding g, see note at A190803 *)
b = (a + 2)/3; c = a/3; r = Range[1, 900];
d = Intersection[b, r](* illustrates closure property *)
e = Intersection[c, r](* illustrates closure property *)
2 FromDigits[#, 3]&/@Tuples[{0, 1}, 7] + 1 (* Vincenzo Librandi, Jul 10 2019 *)

a(n) = 2*A005836(n) + 1. - Charles R Greathouse IV, Sep 06 2011
a(n) = A005823(n) + 1. - Vladimir Shevelev, Dec 17 2012
a(n) = (A191108(n) + 1)/2. - Peter Munn, Jul 09 2019

A173931 Primitive numbers k such that m/k is in the Cantor set for some m.

Original entry on oeis.org

4, 10, 13, 28, 40, 82, 91, 121, 146, 182, 205, 244, 328, 364, 386, 656, 671, 730, 757, 820, 949, 1036, 1093, 1342, 1640, 2044, 2188, 2362, 2555, 2644, 2684, 2812, 2920, 3280, 3640, 3796, 3851, 4088, 4561, 4745, 5110, 6176, 6562, 6643, 7381, 7592, 7913

Offset: 1

T. D. Noe, Mar 03 2010

nonn

Primitive means no k is a multiple of 3. This is sequence A054591 without the multiples of 3. Sequence A173793 is a subsequence. Sequence A173932 gives the least m such for each k. Sequence A173933 gives the number of m < k/2 such that m/k is in the Cantor set. Irregular triangle A173934 gives a row of m values for each k.
The remaining terms <10000 are 9139, 9490, 9841.
It is assumed that gcd(m,k) = 1.

T. D. Noe, Table of n, a(n) for n=1..185

InCantorQ[m_, n_] := !MemberQ[Union[Flatten[RealDigits[m/n,3][[1]]]], 1]; cantor=Reap[Do[If[Mod[n,3] > 0, s=Select[Range[Ceiling[n/2]], GCD[n,# ]==1 && InCantorQ[ #,n] &]; If[s != {}, Sow[{n, s}]]], {n,10000}]][[2,1]]; First[Transpose[cantor]]

A173933 The number of numbers m < k/2 such that m/k is a reduced fraction in the Cantor set, where k= A173931(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 8, 6, 15, 6, 6, 8, 15, 8, 12, 8, 8, 10, 24, 27, 16, 12, 9, 63, 10, 16, 12, 63, 20, 12, 11, 10, 36, 12, 56, 12, 12, 44, 12, 15, 36, 12, 16, 120, 60, 110, 24, 16, 18, 24, 225

Offset: 1

T. D. Noe, Mar 03 2010

nonn

When k is a prime of the form (3^r-1)/2, then the m are 2^r-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.

			When k=40, then 1/k, 3/k, 9/k, and 13/k have base-3 representations containing only the digits 0 and 2.

T. D. Noe, Table of n, a(n) for n = 1..185

Cf. A005823, A005836, A007734, A076481, A173931, A173934.

Length /@ Last[Transpose[cantor]] (* see A173931 *)

Name qualified by Peter Munn, Jul 14 2019

A173932 Least number m such that m/A173931(n) is a reduced fraction in the Cantor set.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 7, 1, 1, 1, 11, 17, 8, 1, 1, 1, 1, 1, 1, 37, 47, 19, 1, 1, 71, 25, 23, 1, 1, 1, 121, 43, 11, 5, 13, 2, 61, 59, 1, 1, 1, 197, 95, 28, 1, 1

Offset: 1

T. D. Noe, Mar 03 2010

nonn

When a(n)=1, then A173931(n) is in sequence A173793.

T. D. Noe, Table of n, a(n) for n=1..185

Cf. A173931, A173793, A173934.

First /@ Last[Transpose[cantor]] (* see A173931 *)

Name qualified by Peter Munn, Jul 12 2019

cp's OEIS Frontend

A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A173931 Primitive numbers k such that m/k is in the Cantor set for some m.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

A173933 The number of numbers m < k/2 such that m/k is a reduced fraction in the Cantor set, where k= A173931(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A173932 Least number m such that m/A173931(n) is a reduced fraction in the Cantor set.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions