cp's OEIS Frontend

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.

%I A242061 #31 May 21 2025 18:46:25
%S A242061 14,2,129,52686,29,7,9,1274,296125969,12012350,5,1279281,44,
%T A242061 302583265614,780914,90,316,2605,106023820090609,1754402265205275806,
%U A242061 7794,72957466300254,768323201,40,18505,23,6478321,3966329,326,14280500082452241
%N 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.
%C A242061 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).
%C A242061 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 cataloged by Hisanori Mishima (see Links).
%H A242061 Lance Fortnow, <a href="http://blog.computationalcomplexity.org/2004/03/counting-rationals-quickly.html">Counting the Rationals Quickly</a>, Computational Complexity Weblog, Monday, March 01, 2004.
%H A242061 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math10/matb2000.htm">361 Congruent Numbers g: 1<=g<=999</a>, Mathematician's Secret Room, Chapter 10 : Congruent Numbers (D27 Congruent numbers).
%H A242061 Yoram Sagher, <a href="http://www.jstor.org/stable/2324846">Counting the rationals</a>, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
%e A242061 .  j       {A226314(n),A054531(n)}, 1<=i<=j<=12 and n=i+j(j-1)/2
%e A242061 .  --   --------------------------------------------------------
%e A242061 .   1:  1,1
%e A242061 .   2:  1,2 2,1
%e A242061 .   3:  1,3 2,3 3,1
%e A242061 .   4:  1,4 3,2 3,4 4,1
%e A242061 .   5:  1,5 2,5 3,5 4,5 5,1
%e A242061 .   6:  1,6 4,3 5,2 5,3 5,6 6,1
%e A242061 .   7:  1,7 2,7 3,7 4,7 5,7 6,7 7,1
%e A242061 .   8:  1,8 5,4 3,8 7,2 5,8 7,4 7,8 8,1
%e A242061 .   9:  1,9 2,9 7,3 4,9 5,9 8,3 7,9 8,9 9,1
%e A242061 .  10:  1,10 6,5 3,10 7,5 9,2 8,5 7,10 9,5 9,10 10,1
%e A242061 .  11:  1,11 2,11 3,11 4,11 5,11 6,11 7,11 8,11 9,11 10,11 11,1
%e A242061 .  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 .
%e A242061 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.
%t A242061 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]
%Y A242061 Cf. A006991, A054531, A226314.
%K A242061 nonn
%O A242061 1,1
%A A242061 _Frank M Jackson_, Aug 13 2014