A038568 - OEIS

%I #39 Feb 25 2021 13:21:33

%S 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,

%T 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,

%U 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

%N Numerators in canonical bijection from positive integers to positive rationals.

%C 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

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

%H David Wasserman, <a href="/A038568/b038568.txt">Table of n, a(n) for n = 0..100000</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Ra#rational">Index entries for sequences related to enumerating the rationals</a>

%e First arrange fractions by increasing denominator, then by increasing numerator:

%e 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, ... (this is A038566/A038567);

%e now follow each term (except the first) with its reciprocal:

%e 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).

%p 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)

%t 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-François Alcover_, Nov 21 2012, translated from Maple *)

%o (Python)

%o from sympy import totient, gcd

%o def a(n):

%o     s=1

%o     k=2

%o     while s<n:

%o         s+=2*totient(k)

%o         k+=1

%o     s-=2*totient(k - 1)

%o     j=1

%o     while s<n:

%o         if gcd(j, k - 1)==1: s+=2

%o         j+=1

%o     if s>n: return j - 1

%o     return k - 1 # _Indranil Ghosh_, May 23 2017, translated from Mathematica

%o (Julia)

%o using Nemo

%o function A038568List(len)

%o     a, A = QQ(0), []

%o     for n in 1:len

%o         a = next_minimal(a)

%o         push!(A, numerator(a))

%o     end

%o A end

%o A038568List(84) |> println # _Peter Luschny_, Mar 13 2018

%o (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 ([p,q][n+2]), n-=2))), n-=2*e)) } \\ _Rémy Sigrist_, Feb 25 2021

%Y Cf. A020652, A020653, A038566, A038567, A038569.

%K nonn,frac,core,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Erich Friedman_