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See Comments at A190803, which is a proper subsequence of A190841.

See Comments at A190803, which is a proper subsequence of A190842.

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then hx+i and jx+k are in a, where h,i,j,k are integers." If m>1, at least one of the numbers b(m)=(a(m)-i)/h and c(m)=(a(m)-k)/j is in the set N of natural numbers. Let d(n) be the n-th b(m) in N, and let e(n) be the n-th c(m) in N. Note that a is a subsequence of both d and e. Examples:

A191113: (h,i,j,k)=(3,-2,4,-2); d=A191146, e=A191149

A191114: (h,i,j,k)=(3,-2,4,-1); d=A191151, e=A191121

A191115: (h,i,j,k)=(3,-2,4,0); d=A191113, e=A191154

A191116: (h,i,j,k)=(3,-2,4,1); d=A191155 e=A191129

A191117: (h,i,j,k)=(3,-2,4,2); d=A191157, e=A191158

A191118: (h,i,j,k)=(3,-2,4,3); d=A191114, e=A191138

...

A191119: (h,i,j,k)=(3,-1,4,-3); d=A191120, e=A191163

A191120: (h,i,j,k)=(3,-1,4,-2); d=A191129, e=A191165

A191121: (h,i,j,k)=(3,-1,4,-1); d=A191166, e=A191167

A191122: (h,i,j,k)=(3,-1,4,0); d=A191168, e=A191169

A191123: (h,i,j,k)=(3,-1,4,1); d=A191170, e=A191171

A191124: (h,i,j,k)=(3,-1,4,2); d=A191172, e=A191173

A191125: (h,i,j,k)=(3,-1,4,3); d=A191174, e=A191175

...

A191126: (h,i,j,k)=(3,0,4,-3); d=A191128, e=A191177

A191127: (h,i,j,k)=(3,0,4,-2); d=A191178, e=A191179

A191128: (h,i,j,k)=(3,0,4,-1); d=A191180, e=A191181

A025613: (h,i,j,k)=(3,0,4,0); d=e=A025613

A191129: (h,i,j,k)=(3,0,4,1); d=A191182, e=A191183

A191130: (h,i,j,k)=(3,0,4,2); d=A191184, e=A191185

A191131: (h,i,j,k)=(3,0,4,3); d=A191186, e=A191187

...

A191132: (h,i,j,k)=(3,1,4,-3); d=A191135, e=A191189

A191133: (h,i,j,k)=(3,1,4,-2); d=A191190, e=A191191

A191134: (h,i,j,k)=(3,1,4,-1); d=A191192, e=A191193

A191135: (h,i,j,k)=(3,1,4,0); d=A191136, e=A191195

A191136: (h,i,j,k)=(3,1,4,1); d=A191196, e=A191197

A191137: (h,i,j,k)=(3,1,4,2); d=A191198, e=A191199

A191138: (h,i,j,k)=(3,1,4,3); d=A191200, e=A191201

...

A191139: (h,i,j,k)=(3,2,4,-3); d=A191143, e=A191119

A191140: (h,i,j,k)=(3,2,4,-2); d=A191204, e=A191205

A191141: (h,i,j,k)=(3,2,4,-1); d=A191206, e=A191207

A191142: (h,i,j,k)=(3,2,4,0); d=A191208, e=A191209

A191143: (h,i,j,k)=(3,2,4,1); d=A191210, e=A191136

A191144: (h,i,j,k)=(3,2,4,2); d=A191212, e=A191213

A191145: (h,i,j,k)=(3,2,4,3); d=e=A191145

...

Representative divisibility properties:

if s=A191116, then 2|(s+1), 4|(s+3), and 8|(s+3) for n>1; if s=A191117, then 10|(s+4) for n>1.

For lists of other "rules sequences" see A190803 (h=2 and j=3) and A191106 (h=j=3).

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

The method generalizes: a finite set F={f} of functions f:N->N and finite set G of numbers generate a set S by these rules: (1) every element of G is in S, and (2) if x is in S then f(x) is in S for every f in F. The sequence a results by taking the numbers in S in increasing order.

Examples include A190803, A191106, A191113, and these:

A191203: 2x, 1+x^2

A191211: 1+2x, 1+x^2

A191281: 2x, x^2-x+1

A191282: 2x, x^2+x+1

A191283: 2x, x(x+1)/2

A191284: floor(3x/2), 2x

A191285: 3x, floor((x^2)/2)

A191286: 3x, 1+x^2

A191287: floor(3x/2), 3x

A191288: 2x, floor((x^2)/3)

A191289: 3x-1, x^2

A191290: 2x+1, x(x+1)/2

For A191203 and other such sequences, the depth g for the NestList in the Mathematica program must be large enough to generate as many terms as required by the user. For example, the rules 2x and 1+x^2, starting with x=1, successively generate set of numbers whose minima are powers of 2: 1->2->4-> ... 2^g -> ....

For general discussions, see A190803 and A191106.

Numbers whose base-3 representation ends in 1 and contains no 2; primitive members of A005836. - Peter Munn, Aug 14 2023

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)

See A190803.

See A190803.

See A190803.