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

Author

Clark Kimberling, Oct 31 2013

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to enumerating the rationals

Crossrefs

Cf. A077610, A229996.

Programs

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 *)

Extensions

Definition corrected by Clark Kimberling, Jun 16 2018