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
(PARI) a(n) = my(s=0, k=1, j=1); while(s<n, s+=eulerphi(k++)); s-=eulerphi(k); while(s<n, if(1==gcd(j, k), s++); j++); k+1-j; \\ Ruud H.G. van Tol, May 14 2024
CROSSREFS
KEYWORD
nonn,frac,core,nice
AUTHOR
EXTENSIONS
Definition clarified by N. J. A. Sloane, Nov 25 2021
STATUS
approved