A190809 -id:A190809 - cp's OEIS frontend

A190853 Integers in (A190809)/2; contains A190809 as a proper subsequence.

1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 20, 22, 25, 26, 28, 31, 32, 34, 40, 43, 44, 49, 50, 52, 53, 56, 61, 62, 64, 67, 68, 76, 79, 80, 85, 86, 88, 94, 97, 98, 100, 103, 104, 106, 112, 121, 122, 124, 128, 130, 133, 134, 136, 148, 151, 152, 157, 158, 160, 161, 169, 170, 172, 176, 184, 187, 188, 193, 194, 196, 200, 202, 205, 206, 208, 212

Offset: 1

Clark Kimberling, May 25 2011

nonn

See A190803.

Cf. A190803, A190809, A190854.

Mathematica
```
(See A190809.)
```

A190854 Integers in (-2+A190809)/3; contains A190809 as a proper subsequence.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 14, 16, 17, 18, 20, 22, 26, 28, 32, 34, 40, 42, 44, 50, 52, 53, 56, 58, 62, 64, 66, 68, 70, 74, 80, 82, 86, 88, 90, 98, 100, 104, 106, 112, 114, 122, 124, 128, 130, 134, 136, 138, 152, 158, 160, 161, 162, 170, 172, 176, 178, 188, 194, 196, 200, 202, 206, 208, 210, 212, 214, 224, 226, 234, 242, 244, 248, 250, 256

Offset: 1

Clark Kimberling, May 25 2011

nonn

See A190803.

Cf. A190803, A190909, A190853.

Mathematica
```
(See A190809.)
```

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 14, 15, 17, 23, 26, 27, 29, 33, 41, 44, 45, 50, 51, 53, 57, 65, 68, 77, 80, 81, 86, 87, 89, 98, 99, 101, 105, 113, 122, 129, 131, 134, 135, 149, 152, 153, 158, 159, 161, 170, 171, 173, 177, 194, 195, 197, 201, 203, 209, 225, 230, 239, 242

Offset: 1

Clark Kimberling, May 25 2011

nonn

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, where [A......] indicates a conjecture:
A190803: (h,i,j,k)=(2,-1,3,-1); d=A190841, e=A190842
A190804: (h,i,j,k)=(2,-1,3,0); d=[A190803], e=A190844
A190805: (h,i,j,k)=(2,-1,3,1); d=A190845, e=[A190808]
A190806: (h,i,j,k)=(2,-1,3,2); d=[A190804], e=A190848
...
A190807: (h,i,j,k)=(2,0,3,-1); d=A190849, e=A190850
A003586: (h,i,j,k)=(2,0,3,0); d=e=A003586
A190808: (h,i,j,k)=(2,0,3,1); d=A190851, e=A190852
A190809: (h,i,j,k)=(2,0,3,2); d=A190853, e=A190854
...
A190810: (h,i,j,k)=(2,1,3,-1); d=A190855, e=A190856
A190811: (h,i,j,k)=(2,1,3,0); d=A002977, e=A190857
A002977: (h,i,j,k)=(2,1,3,1); d=A190858, e=A190859
A190812: (h,i,j,k)=(2,1,3,2); d=A069353, e=[A190812]
...
For h=j=3, see A191106; for h=3 and j=4, see A191113.

			1 -> 2 -> 3,5 -> 8,9,14 -> 15,17,23,26,27,41 -> ...

Reinhard Zumkeller, 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.

Cf. A002977, A003586, A190804-A190812, A190841-A190860.

import Data.Set (singleton, deleteFindMin, insert)
a190803 n = a190803_list !! (n-1)
a190803_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)
b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];
d = Intersection[b, r] (* A190841 *)
e = Intersection[c, r] (* A190842 *)
(* Regarding this program - useful for many choices of h,i,j,k,f,g - the depth g must be chosen with care - that is, large enough.  Assuming that h<=j, the least new terms in successive nests a are typically iterates of hx+i, starting from x=1.  If, for example, h=2 and i=0, the least terms are 2,4,8,...,2^g, so that g>=9 ensures inclusion of all the desired terms <=256. *)

a(34)=225 inserted by Reinhard Zumkeller, Jun 01 2011

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A190853 Integers in (A190809)/2; contains A190809 as a proper subsequence.

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A190854 Integers in (-2+A190809)/3; contains A190809 as a proper subsequence.

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A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

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