OFFSET
1,2
COMMENTS
Starting with g(1) = (1), write the numbers in the ordered union of g(1), g(2),…, g(n) as (x(1),x(2),…,x(m)). Then for i = 1..m, write x(i)/(1 + x(j)) for j = 1..m, and expel all the numbers that have previously occurred. The result is ordered union of g(1), g(2),..., g(n+1). The cardinalities of the first 7 unions are 1, 2, 5, 20, 245, 38179, 1032578826.
Conjecture: every rational number in the interval (0,1] occurs exactly once in T.
EXAMPLE
Successive generations:
g(1) = (1)
g(2) = (1/2)
g(3) = (2/3, 1/4, 1/3)
g(4) = (3/5, 4/5, 3/4, 3/10, 2/5, 3/8, 4/9, 8/15, 1/8, 1/6, 3/20, 1/5, 3/16, 2/9, 4/15)
Let U(n) = ordered union of g(1), g(2), ..., g(n).
U(1) = (1)
U(2) = (1, 1/2)
U(3) = (1, 1/2, 2/3, 1/4, 1/3)
U(4) = (1, 1/2, 2/3, 1/4, 1/3, 3/5, 4/5, 3/4, 3/10, 2/5, 3/8, 4/9, 8/15, 1/8, 1/6, 3/20, 1/5, 3/16, 2/9, 4/15)
Denominators U(4): 1, 2, 3, 4, 3, 5, 5, 4, 10, 5, 8, 9, 15, 8, 6, 20, 5, 16, 9, 15
MATHEMATICA
(* In the remarks below, U(n) = ordered union of generations g(1), g(2), ...g(n) *)
x = {1};
x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(2) *)
x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(3) *)
x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(4) *)
Numerator[x] (* A371279 *)
Denominator[x] (* this sequence *)
(* Peter J. C. Moses, Mar 16 2024 *)
CROSSREFS
KEYWORD
nonn,tabf,frac
AUTHOR
Clark Kimberling, Mar 18 2024
STATUS
approved