A038569 - OEIS

%I #46 Nov 25 2021 12:35:45

%S 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,

%T 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,

%U 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

%N Denominators in a certain bijection from positive integers to positive rationals.

%C 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

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

%H David Wasserman, <a href="/A038569/b038569.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 the positive fractions p/q <= 1 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 but the first term by 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): 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)

%t 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 *)

%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 + 1: return k - 1

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

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

%Y Cf. A038566, A038567, A038568.

%Y See A020652, A020653 for an alternative version.

%K nonn,frac,core,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Erich Friedman_

%E Definition clarified by _N. J. A. Sloane_, Nov 25 2021