A156684 The number of primitive Pythagorean triples A^2+B^2=C^2 with 0 < A < B < C and gcd(A,B)=1, and both legs less than n.

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16

Offset

1,13

Comments

For large N, Benito and Varona have shown that a(N)~2/pi^2 Log(1+sqrt(2)).N +O(sqrt(N)). However, the approximations to a(N)/N are considerably more accurate than the error term suggests, and it certainly appears that the density of the primitive triples with both legs less than N tends towards 2/pi^2 Log(1+sqrt(2))=0.1786... as N becomes large.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Manuel Benito and Juan L. Varona, Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics 143, (2002), pp. 117-126.

Example

There are two primitive triples with both legs less than 14, specifically (3,4,5) and (5,12,13). Hence a(14)=2.

Mathematica

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3, 4, 5}}, i=4, j=5}, While[i<n, If[GCD[i, j]==1, h=Sqrt[i^2+j^2]; If[IntegerQ[h] && j<n, AppendTo[t, {i, j, h}]]; ]; If[j<n, j+=2, i++; j=i+1]]; t]; Append[{0, 0, 0, 0}, Length[PrimitivePythagoreanTriplets[ # ]]&/@Range[5, 50]]//Flatten
(* Second program: *)
Join[{0}, Cases[Import["https://oeis.org/A024360/b024360.txt", "Table"], {_, _}][[;; 10000, 2]]] // Accumulate (* Jean-François Alcover, Mar 27 2020 *)

Crossrefs

Cf. Essentially partial sums of A024360.

Sequence in context: A360010 A276611 A238598 * A070564 A072358 A074795

Adjacent sequences: A156681 A156682 A156683 * A156685 A156686 A156687

Keyword

easy,nice,nonn

Author

Ant King, Feb 17 2009

Status

approved