A371279 Irregular triangular array of numerators of the set T of fractions generated by these rules: g(1) = (1), and if x and y are in T, then x/(y+1) is in T; see Comments.

1, 1, 2, 1, 1, 3, 4, 3, 3, 2, 3, 4, 8, 1, 1, 3, 1, 3, 2, 4, 5, 5, 4, 10, 5, 8, 9, 15, 8, 6, 20, 5, 16, 9, 15, 5, 5, 2, 5, 5, 4, 9, 15, 3, 10, 5, 8, 9, 15, 10, 8, 20, 10, 16, 6, 16, 40, 32, 6, 10, 5, 5, 1, 5, 5, 2, 9, 15, 3, 5, 5, 4, 9, 15, 5, 4, 10, 5, 8, 3

Offset

1,3

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.

Links

Table of n, a(n) for n=1..80.

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)
Numerators in U(4): 1, 1, 2, 1, 1, 3, 4, 3, 3, 2, 3, 4, 8, 1, 1, 3, 1, 3, 2, 4.

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]  (* this sequence *)
Denominator[x]  (* A371280 *)
(* Peter J. C. Moses, Mar 16 2024 *)

Crossrefs

Cf. A226080, A371280.

Sequence in context: A319516 A015138 A157807 * A100529 A262953 A226209

Adjacent sequences: A371276 A371277 A371278 * A371280 A371281 A371282

Keyword

nonn,tabf,frac

Author

Clark Kimberling, Mar 18 2024

Status

approved