A229994 For every positive integer m, let u(m) = (d(1),d(2),...,d(k)) be the unitary divisors of m in increasing order. Let Q be the concatenation of the vectors (d(k)/d(1), d(k-1)/d(2), ..., d(1)/d(k)), so that every positive rational number appears in Q exactly once. The numerators form A229994; the denominators, A077610.

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 3, 2, 1, 7, 1, 8, 1, 9, 1, 10, 5, 2, 1, 11, 1, 12, 4, 3, 1, 13, 1, 14, 7, 2, 1, 15, 5, 3, 1, 16, 1, 17, 1, 18, 9, 2, 1, 19, 1, 20, 5, 4, 1, 21, 7, 3, 1, 22, 11, 2, 1, 23, 1, 24, 8, 3, 1, 25, 1, 26, 13, 2, 1, 27, 1, 28, 7, 4, 1

Offset

1,2

Comments

The number of terms in S(m) is A034444(m); the denominators are given by A077610.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for sequences related to enumerating the rationals

Example

The first fifteen positive rationals:  1, 2, 1/2, 3, 1/3, 4, 1/4, 5, 1/5, 6, 3/2, 2/3, 1/6, 7, 1/7.

Mathematica

z = 40; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; k[n_] := Length[r[n]]; t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}]; u[1] = t[1]; u[n_] := Join[u[n - 1], t[n]];
Numerator[u[z]]   (* A229994 *)
Denominator[u[z]] (* A077610 *)

Crossrefs

Cf. A077610, A229996.

Sequence in context: A308073 A090331 A338759 * A165818 A369895 A194746

Adjacent sequences: A229991 A229992 A229993 * A229995 A229996 A229997

Keyword

nonn,easy

Author

Clark Kimberling, Oct 31 2013

Extensions

Definition corrected by Clark Kimberling, Jun 16 2018

Status

approved