A242061 Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1, x+y odd and for the least y, {x, y} are integers such that x*y(y^2-x^2)/A006991(n) is a perfect square.

14, 2, 129, 52686, 29, 7, 9, 1274, 296125969, 12012350, 5, 1279281, 44, 302583265614, 780914, 90, 316, 2605, 106023820090609, 1754402265205275806, 7794, 72957466300254, 768323201, 40, 18505, 23, 6478321, 3966329, 326, 14280500082452241

Offset

1,1

Comments

The triangle array A226314(n)/A054531(n) that enumerates all positive rationals x/y can be generalized to enumerate all ordered pairs {x, y} where x and y are natural numbers. For example, A243808 uses a subset of this triangular array to enumerate all primitive Pythagorean triples (PPT).

A006991(n) is the sequence of primitive congruent numbers and by definition there exists a PPT whose area is equal to k^2*A006991(n) for some integer k. a(n) is an enumeration of these PPT's and is a measure of the number of Pythagorean triangles that have to be searched to find a PPT with the least hypotenuse that has an area equal to k^2*A006991(n). If {x, y} are the generators of a PPT (a, b, c) where a = y^2-x^2, b = 2x*y, c=y^2+x^2 then its area = x*y(y^2-x^2). The Mathematica program limits searches to the first 12.5 million generated PPT's. All other results have been obtained from tables catalogued by Hisanori Mishima (see Links).

Links

Table of n, a(n) for n=1..30.

Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.

Hisanori Mishima, 361 Congruent Numbers g: 1<=g<=999, Mathematician's Secret Room, Chapter 10 : Congruent Numbers (D27 Congruent numbers).

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(13)=44 and A006991(13)=34 so 34 is the 13th congruent number. a(13) gives the 44th term of the triangular array at index (8, 9). This gives (x, y) as (8, 9), it generates the PPT (17, 144, 145) and has an area 6^2*34 = 1224.

Mathematica

lst1={5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 34, 37, 38, 39, 41, 46, 47, 53, 55, 61, 62, 65, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95, 101}; getpos[n0_] := (lst=0; Do[If[IntegerQ@Sqrt[m*n(m-n)(m+n)/n0]&&OddQ[m+n] && GCD[m, n]==1, (lst=m(m-1)/2+n; Break[])], {m, 2, 5000}, {n, 1, m-1}]; lst); SetAttributes[getpos, Listable]; getpos[lst1]

Crossrefs

Cf. A006991, A054531, A226314.

Sequence in context: A335701 A377037 A303380 * A040189 A088576 A204159

Adjacent sequences: A242058 A242059 A242060 * A242062 A242063 A242064

Keyword

nonn

Author

Frank M Jackson, Aug 13 2014

Status

approved