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

Author

Ant King, Feb 17 2009

Keywords

• easy
• nice
• nonn

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.

Examples

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

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.

Crossrefs

Cf. Essentially partial sums of A024360.

Programs

Mathematica

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[iA024360/b024360.txt", "Table"], {, }][[;; 10000, 2]]] // Accumulate (* Jean-François Alcover, Mar 27 2020 *)