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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 April 8 14:15 EDT 2020. Contains 333314 sequences. (Running on oeis4.)