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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Mar 27 2020 *)

CROSSREFS

Cf. Essentially partial sums of A024360.

Sequence in context: A004257 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

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Last modified April 15 13:37 EDT 2021. Contains 342977 sequences. (Running on oeis4.)