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A038568 Numerators in canonical bijection from positive integers to positive rationals. 7
1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 4, 1, 5, 2, 5, 3, 5, 4, 5, 1, 6, 5, 6, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 1, 8, 3, 8, 5, 8, 7, 8, 1, 9, 2, 9, 4, 9, 5, 9, 7, 9, 8, 9, 1, 10, 3, 10, 7, 10, 9, 10, 1, 11, 2, 11, 3, 11, 4, 11, 5, 11, 6, 11, 7, 11, 8, 11, 9, 11, 10, 11, 1, 12, 5, 12, 7, 12, 11, 12, 1, 13, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Even-indexed terms are positive integers in order, with m occurring phi(m) times. Preceding odd-indexed terms (except for missing initial 0) are the corresponding numbers <= m and relatively prime to m, in increasing order. The denominators are just this sequence shifted left. Thus each positive rational occurs exactly once as a ratio a(n)/a(n+1). - Franklin T. Adams-Watters, Dec 06 2006

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

EXAMPLE

First arrange fractions 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 term with 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): A038568 := 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 (j-1) fi: RETURN (k-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com)

MATHEMATICA

a[n_] := Module[{sum = 1, k = 2}, While[sum < n, sum = sum + 2*EulerPhi[k]; k = k+1]; sum = sum - 2*EulerPhi[k-1]; j = 1; While[sum < n, If[GCD[j, k-1] == 1, sum = sum+2]; j = j+1; ]; If[sum > n, Return[j-1]]; Return[k-1] ]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, Nov 21 2012, translated from 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: return j - 1

    return k - 1 # Indranil Ghosh, May 23 2017, translated from Mathematica

(Julia)

using Nemo

function A038568List(len)

    a, A = QQ(0), []

    for n in 1:len

        a = next_minimal(a)

        push!(A, numerator(a))

    end

A end

A038568List(84) |> println # Peter Luschny, Mar 13 2018

CROSSREFS

Cf. A020652, A020653, A038566-A038569.

Sequence in context: A215467 A284266 A317988 * A071912 A070940 A020651

Adjacent sequences:  A038565 A038566 A038567 * A038569 A038570 A038571

KEYWORD

nonn,frac,core,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Erich Friedman

STATUS

approved

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Last modified November 14 03:52 EST 2018. Contains 317159 sequences. (Running on oeis4.)