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 A243808 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1 and x+y odd. 4
 2, 5, 7, 9, 12, 14, 16, 20, 23, 25, 27, 29, 31, 33, 35, 38, 40, 44, 46, 48, 52, 54, 57, 59, 61, 63, 65, 67, 71, 73, 77, 80, 82, 84, 86, 88, 90, 92, 94, 96, 100, 102, 104, 107, 109, 113, 119, 121, 123, 125, 127, 129, 131, 133, 135, 138, 140 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS If {x,y} are used as the generators of primitive Pythagorean triples (a,b,c) where a=y^2-x^2, b=2x*y and c=x^2+y^2, then the above sequence enumerates all PPT's as a 1-to-1 mapping into the integers. LINKS Frank M Jackson, Table of n, a(n) for n = 1..57 Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004. Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001. EXAMPLE .  j       {A226314(n),A054531(n)}, 1<=i<=j<=12 and n=i+j(j-1)/2 .  --   ------------------------------------------------------------ .   1:  1,1 .   2:  1,2 2,1 .   3:  1,3 2,3 3,1 .   4:  1,4 3,2 3,4 4,1 .   5:  1,5 2,5 3,5 4,5 5,1 .   6:  1,6 4,3 5,2 5,3 5,6 6,1 .   7:  1,7 2,7 3,7 4,7 5,7 6,7 7,1 .   8:  1,8 5,4 3,8 7,2 5,8 7,4 7,8 8,1 .   9:  1,9 2,9 7,3 4,9 5,9 8,3 7,9 8,9 9,1 .  10:  1,10 6,5 3,10 7,5 9,2 8,5 7,10 9,5 9,10 10,1 .  11:  1,11 2,11 3,11 4,11 5,11 6,11 7,11 8,11 9,11 10,11 11,1 .  12:  1,12 7,6 9,4 10,3 5,12 11,2 7,12 11,3 11,4 11,6 11,12 12,1 . a(4)=9, as the 4th PPT is generated from the 9th term of the triangular array at index (3,4). This gives (x,y) as (3,4) and it generates the PPT (7,24,25). Conversely the PPT (7,24,25) gives (x,y) = (sqrt((25-7)/2), sqrt((25+7)/2)=(3,4). It is the 9th term of the triangular array and the 4th term of the enumerating sequence. MATHEMATICA ratmap[p_, q_] := (q(q-1)/2+p); mm=20; lst={}; Do[If[OddQ[m+n]&&GCD[m, n]==1, AppendTo[lst, n/m]], {m, 1, mm}, {n, 1, m}]; Sort@Table[ratmap[Numerator[lst[[k]]], Denominator[lst[[k]]]], {k, 1, Length[lst]}] CROSSREFS Cf. A054531, A169581, A226314. Sequence in context: A184749 A047386 A187416 * A275278 A184007 A003256 Adjacent sequences:  A243805 A243806 A243807 * A243809 A243810 A243811 KEYWORD nonn AUTHOR Frank M Jackson, Jun 13 2014 STATUS approved

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Last modified March 30 06:59 EDT 2020. Contains 333119 sequences. (Running on oeis4.)