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A020653 Denominators in a certain bijection from positive integers to positive rationals. 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
REFERENCES
Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.
H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.
LINKS
EXAMPLE
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)
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 *)
PROG
(Haskell)
a020653 n = a020653_list !! (n-1)
a020653_list = concat $ map reverse $ tail a038566_tabf
-- Reinhard Zumkeller, Oct 30 2012
(Python)
from sympy import totient, gcd
def a(n):
s=0
k=2
while s<n:
s+=totient(k)
k+=1
s-=totient(k - 1)
j=1
while s<n:
if gcd(j, k - 1)==1: s+=1
j+=1
return k - j # Indranil Ghosh, May 23 2017, translated from Ulrich Schimke's MAPLE code
CROSSREFS
Sequence in context: A088445 A280696 A293247 * A094522 A233204 A308058
KEYWORD
nonn,frac,core,nice
AUTHOR
EXTENSIONS
Definition clarified by N. J. A. Sloane, Nov 25 2021
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)