A020653 Denominators in a certain bijection from positive integers to positive rationals.

1, 2, 1, 3, 1, 4, 3, 2, 1, 5, 1, 6, 5, 4, 3, 2, 1, 7, 5, 3, 1, 8, 7, 5, 4, 2, 1, 9, 7, 3, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 7, 5, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 11, 9, 5, 3, 1, 14, 13, 11, 8, 7, 4, 2, 1, 15, 13, 11, 9, 7, 5, 3, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• frac
• core
• nice

Comments

This bijection lists the fractions p/q (in lowest terms) by increasing p+q, then by increasing p (see the example). The variant A038569 corresponds to the bijection where each fraction p/q with p < q is followed by its reciprocal q/p. - M. F. Hasler, Oct 25 2021

Examples

			From _M. F. Hasler_, Nov 25 2021: (Start)
This sequence gives the denominators of the positive fractions p/q (in lowest terms) when they are listed by increasing p+q, then by increasing p:
1/1; 1/2, 2/1; 1/3, 3/1; 1/4, 2/3, 3/2, 4/1; 1/5, 5/1; 1/6, 2/5, 3/4, 4/3, 5/2, 6/1; ...
(End)
		

References

• Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.
• H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.

Links

• David Wasserman, Table of n, a(n) for n = 1..100000
• Index entries for "core" sequences
• Index entries for sequences related to enumerating the rationals

Crossrefs

Cf. A020652, A038566, A038569.

Programs

Haskell

a020653 n = a020653_list !! (n-1)
a020653_list = concat $ map reverse $ tail a038566_tabf
-- Reinhard Zumkeller, Oct 30 2012

Maple

with (numtheory): A020653 := proc (n) local sum, j, k; sum := 0: k := 2: while (sum < n) do: sum := sum + phi(k): k := k + 1: od: sum := sum - phi(k-1): j := 1; while sum < n do: if gcd(j,k-1) = 1 then sum := sum + 1: fi: j := j+1: od: RETURN (k-j): end: # Ulrich Schimke (ulrschimke(AT)aol.com), Nov 06 2001

Mathematica

a[n_] := Module[{s=0, k=2}, While [s < n, s = s + EulerPhi[k]; k = k+1]; s = s - EulerPhi[k-1]; j=1; While[s < n , If[GCD[j, k-1] == 1 , s = s+1]; j = j+1]; k-j]; Table[a[n], {n, 1, 96}] (* Jean-François Alcover, Dec 06 2012, after Ulrich Schimke's Maple program *)
Flatten[Map[Denominator[#/Reverse[#]]&,Table[Flatten[Position[GCD[Map[Mod[#,n]&,Range[n-1]],n],1]],{n,100}]]] (* Peter J. C. Moses, Apr 17 2013 *)

PARI

a(n) = my(s=0, k=1, j=1); while(sRuud H.G. van Tol, May 14 2024

Python

from sympy import totient, gcd
def a(n):
    s=0
    k=2
    while sIndranil Ghosh, May 23 2017, translated from Ulrich Schimke's MAPLE code

Extensions

Definition clarified by N. J. A. Sloane, Nov 25 2021