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A038569
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Denominators in a certain bijection from positive integers to positive rationals.
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13
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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, 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, 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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.
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LINKS
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EXAMPLE
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First arrange the positive fractions p/q <= 1 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 but the first term by 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).
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MAPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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 + 1: return k - 1
return j - 1 # Indranil Ghosh, May 23 2017, translated from Mathematica
(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
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CROSSREFS
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KEYWORD
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nonn,frac,core,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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