A038569 Denominators in a certain bijection from positive integers to positive rationals.

1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 3, 5, 1, 5, 2, 5, 3, 5, 4, 6, 1, 6, 5, 7, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 8, 1, 8, 3, 8, 5, 8, 7, 9, 1, 9, 2, 9, 4, 9, 5, 9, 7, 9, 8, 10, 1, 10, 3, 10, 7, 10, 9, 11, 1, 11, 2, 11, 3, 11, 4, 11, 5, 11, 6, 11, 7, 11, 8, 11, 9, 11, 10, 12, 1, 12, 5, 12, 7, 12, 11, 13, 1, 13

Offset

0,2

Comments

See A020652/A020653 for an alternative version where the fractions p/q are listed by increasing p+q, then p. - M. F. Hasler, Nov 25 2021

References

H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.

Links

David Wasserman, Table of n, a(n) for n = 0..100000

Index entries for "core" sequences

Index entries for sequences related to enumerating the rationals

Example

First arrange the positive fractions p/q <= 1 by increasing denominator, then by increasing numerator:
1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ... (this is A038566/A038567).
Now follow each but the first term by its reciprocal:
1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, ... (this is A038568/A038569).

Maple

with (numtheory): A038569 := proc (n) local sum, j, k; sum := 1: k := 2: while (sum < n) do: sum := sum + 2 * phi(k): k := k + 1: od: sum := sum - 2 * phi(k-1): j := 1: while sum < n do: if gcd(j, k-1) = 1 then sum := sum + 2: fi: j := j+1: od: if sum > n then RETURN (k-1) fi: RETURN (j-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)

Mathematica

a[n_] := Module[{s = 1, k = 2, j = 1}, While[s <= n, s = s + 2*EulerPhi[k]; k = k+1]; s = s - 2*EulerPhi[k-1]; While[s <= n, If[GCD[j, k-1] == 1, s = s+2]; j = j+1]; If[s > n+1, k-1, j-1]]; Table[a[n], {n, 0, 99}](* Jean-François Alcover, Nov 10 2011, after Maple *)

Prog

(Python)
from sympy import totient, gcd
def a(n):
    s=1
    k=2
    while s<=n:
        s+=2*totient(k)
        k+=1
    s-=2*totient(k - 1)
    j=1
    while s<=n:
        if gcd(j, k - 1)==1: s+=2
        j+=1
    if s>n + 1: return k - 1
    return j - 1 # Indranil Ghosh, May 23 2017, translated from Mathematica
(PARI) a(n) = { my (e); for (q=1, oo, if (n+1<2*e=eulerphi(q), for (p=1, oo, if (gcd(p, q)==1, if (n+1<2, return ([q, p][n+2]), n-=2))), n-=2*e)) } \\ Rémy Sigrist, Feb 25 2021

Crossrefs

Cf. A038566, A038567, A038568.

See A020652, A020653 for an alternative version.

Sequence in context: A366737 A316436 A303674 * A308686 A020650 A361373

Adjacent sequences: A038566 A038567 A038568 * A038570 A038571 A038572

Keyword

nonn,frac,core,nice

Author

N. J. A. Sloane

Extensions

More terms from Erich Friedman

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

Status

approved