

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:
...
...
...


LINKS



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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



