

A190803


Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x1 and 3x1 are in a.


32



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
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OFFSET

1,2


COMMENTS

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 nth b(m) in N, and let e(n) be the nth 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.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely SelfGenerating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 113.


EXAMPLE

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


MATHEMATICA

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. *)


PROG

(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a190803 n = a190803_list !! (n1)
a190803_list = 1 : f (singleton 2)
where f s = m : (f $ insert (2*m1) $ insert (3*m1) s')
where (m, s') = deleteFindMin s
 Reinhard Zumkeller, Jun 01 2011


CROSSREFS

Cf. A002977, A003586, A190804A190812, A190841A190860.
Sequence in context: A058237 A251599 A009388 * A125871 A141399 A104737
Adjacent sequences: A190800 A190801 A190802 * A190804 A190805 A190806


KEYWORD

nonn


AUTHOR

Clark Kimberling, May 25 2011


EXTENSIONS

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


STATUS

approved



