A046079 Number of Pythagorean triangles with leg n.

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 3, 1, 2, 1, 4, 4, 1, 1, 7, 2, 1, 3, 4, 1, 4, 1, 4, 4, 1, 4, 7, 1, 1, 4, 7, 1, 4, 1, 4, 7, 1, 1, 10, 2, 2, 4, 4, 1, 3, 4, 7, 4, 1, 1, 13, 1, 1, 7, 5, 4, 4, 1, 4, 4, 4, 1, 12, 1, 1, 7, 4, 4, 4, 1, 10, 4, 1, 1, 13, 4, 1, 4, 7, 1, 7, 4, 4, 4, 1, 4, 13, 1, 2, 7

Offset

1,8

Comments

Number of ways in which n can be the leg (other than the hypotenuse) of a primitive or nonprimitive right triangle.

Number of ways that 2/n can be written as a sum of exactly two distinct unit fractions. For every solution to 2/n = 1/x + 1/y, x < y, the Pythagorean triple is (n, y-x, x+y-n). - T. D. Noe, Sep 11 2002

For n>2, the positions of the ones in this sequence correspond to the prime numbers and their doubles, A001751. - Ant King, Jan 29 2011

Let L = length of longest leg, H = hypotenuse. For odd n: L =(n^2-1)/2 and H = L+1. For even n, L = (n^2-4)/4 and H = L+2. - Richard R. Forberg, May 31 2013

Or number of ways n^2 can be written as the difference of two positive squares: a(3) = 1: 3^2 = 5^2-4^2; a(8) = 2: 8^2 = 10^2-6^2 = 17^2-15^2; a(16) = 3: 16^2 = 20^2-12^2 = 34^2-30^2 = 65^2-63^2. - Alois P. Heinz, Aug 06 2019

Number of ways to write 2n as the sum of two positive integers r and s such that r < s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 21 2020

References

A. Beiler, Recreations in the Theory of Numbers. New York: Dover Publications, pp. 116-117, 1966.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Hanz Becker, Pythagorean triples in JavaScript

Ron Knott, Pythagorean Triples and Online Calculators

Project Euler, Problem 176: Right-angled triangles that share a cathetus

F. Richman, Pythagorean Triples

A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 6.

Eric Weisstein's World of Mathematics, Pythagorean Triple

Formula

For odd n, a(n) = A018892(n) - 1.

Let n = (2^a0)*(p1^a1)*...*(pk^ak). Then a(n) = [(2*a0 - 1)*(2*a1 + 1)*(2*a2 + 1)*(2*a3 + 1)*...*(2*ak + 1) - 1]/2. Note that if there is no a0 term, i.e., if n is odd, then the first term is simply omitted. - Temple Keller (temple.keller(AT)gmail.com), Jan 05 2008

For odd n, a(n) = (tau(n^2) - 1) / 2; for even n, a(n) = (tau((n / 2)^2) - 1) / 2. - Amber Hu (hupo001(AT)gmail.com), Jan 23 2008

a(n) = Sum_{i=1..n-1} (1 - ceiling(i*(2*n-i)/(2*n-2*i)) + floor(i*(2*n-i)/(2*n-2*i))). - Wesley Ivan Hurt, Apr 21 2020

Mathematica

a[n_] := (DivisorSigma[0, If[OddQ[n], n, n / 2]^2] - 1) / 2; Table[a[i], {i, 100}] (* Amber Hu (hupo001(AT)gmail.com), Jan 23 2008 *)
a[ n_] := Length @ FindInstance[ n > 0 && y > 0 && z > 0 && n^2 + y^2 == z^2, {y, z}, Integers, 10^9]; (* Michael Somos, Jul 25 2018 *)

Prog

(Sage) def A046079(n) : return (number_of_divisors(n^2 if n%2==1 else n^2/4) - 1) // 2 # Eric M. Schmidt, Jan 26 2013
(PARI) A046079(n) = ((numdiv(if(n%2, n, n/2)^2)-1)/2); \\ Antti Karttunen, Sep 27 2018
(Python)
from math import prod
from sympy import factorint
def A046079(n): return prod((e+(p&1)<<1)-1 for p, e in factorint(n).items())>>1 # Chai Wah Wu, Sep 06 2022

Crossrefs

Cf. A000290, A046080, A046081, A001227, A018892, A024361, A024362, A024363.

Sequence in context: A273432 A284343 A033151 * A319700 A279104 A165509

Adjacent sequences: A046076 A046077 A046078 * A046080 A046081 A046082

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved