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A020652 Numerators in canonical bijection from positive integers to positive rationals.

%I A020652 #60 May 12 2024 13:08:29
%S A020652 1,1,2,1,3,1,2,3,4,1,5,1,2,3,4,5,6,1,3,5,7,1,2,4,5,7,8,1,3,7,9,1,2,3,
%T A020652 4,5,6,7,8,9,10,1,5,7,11,1,2,3,4,5,6,7,8,9,10,11,12,1,3,5,9,11,13,1,2,
%U A020652 4,7,8,11,13,14,1,3,5,7,9,11,13,15,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,5
%N A020652 Numerators in canonical bijection from positive integers to positive rationals.
%C A020652 a(A002088(n)) = 1 for n > 1. - _Reinhard Zumkeller_, Jul 29 2012
%C A020652 When read as an irregular table with each 1 entry starting a new row, then the n-th row consists of the set of multiplicative units of Z_{n+1}. These rows form a group under multiplication mod n. - _Tom Edgar_, Aug 20 2013
%C A020652 The pair of sequences A020652/A020653 is defined by ordering the positive fractions p/q (reduced to lowest terms) by increasing p+q, then increasing p: 1/1; 1/2, 2/1; 1/3, 3/1; 1/4, 2/3, 3/2, 4/1; 1/5, 5/1; 2/5, 3/4, 4/3, 5/2; etc. For given p+q, there are A000010(p+q) fractions, listed starting at index A002088(p+q-1). - _M. F. Hasler_, Mar 06 2020
%D A020652 S. Cook, Problem 511: An Enumeration Problem, Journal of Recreational Mathematics, Vol. 9:2 (1976-77), 137. Solution by the Problem Editor, JRM, Vol. 10:2 (1977-78), 122-123.
%D A020652 Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.
%D A020652 H. Lauwerier, Fractals, Princeton Univ. Press, p. 23.
%H A020652 David Wasserman, <a href="/A020652/b020652.txt">Table of n, a(n) for n = 1..100000</a>
%H A020652 Paul Yiu, <a href="http://math.fau.edu/Yiu/RecreationalMathematics2003.pdf">Recreational Mathematics</a>, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.
%H A020652 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A020652 <a href="/index/Ra#rational">Index entries for sequences related to enumerating the rationals</a>
%H A020652 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%e A020652 Arrange positive fractions < 1 by increasing denominator then by increasing numerator: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6 ... (this is A020652/A038567). - _William Rex Marshall_, Dec 16 2010
%p A020652 with (numtheory): A020652 := 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 (j-1): end: # Ulrich Schimke (ulrschimke(AT)aol.com), Nov 06 2001
%t A020652 Reap[Do[If[GCD[num, den] == 1, Sow[num]], {den, 1, 20}, {num, 1, den-1}] ][[2, 1]] (* _Jean-François Alcover_, Oct 22 2012 *)
%o A020652 (Haskell)
%o A020652 a020652 n = a020652_list !! (n-1)
%o A020652 a020652_list = map fst [(u,v) | v <- [1..], u <- [1..v-1], gcd u v == 1]
%o A020652 -- _Reinhard Zumkeller_, Jul 29 2012
%o A020652 (PARI) a(n)=my(s,j=1,k=1);while(s<n,s+=eulerphi(k++););s-=eulerphi(k);while(s<n,if(gcd(j,k)==1,s++);j++);j-1 \\ _Charles R Greathouse IV_, Feb 07 2013
%o A020652 (Python)
%o A020652 from sympy import totient, gcd
%o A020652 def a(n):
%o A020652     s=0
%o A020652     k=2
%o A020652     while s<n:
%o A020652         s+=totient(k)
%o A020652         k+=1
%o A020652     s-=totient(k - 1)
%o A020652     j=1
%o A020652     while s<n:
%o A020652         if gcd(j, k - 1)==1:
%o A020652             s+=1
%o A020652         j+=1
%o A020652     return j - 1
%o A020652 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, May 23 2017, after Ulrich Schimke's MAPLE code
%Y A020652 Essentially the same as A038566, which is the main entry for this sequence.
%Y A020652 Cf. A020653, A038567-A038569, A182972-A182976.
%Y A020652 A054424 gives mapping to Stern-Brocot tree.
%Y A020652 Cf. A037161.
%K A020652 nonn,frac,core,nice,tabf
%O A020652 1,3
%A A020652 _David W. Wilson_