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.

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

Offset: 1

Author

Frank M Jackson, Jun 13 2014

Keywords

• nonn

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.

Examples

			.  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.
		

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.

Crossrefs

Cf. A054531, A169581, A226314.

Programs

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]}]