

A046079


Number of Pythagorean triangles with leg n.


21



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
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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, yx, x+yn).  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^21)/2 and H = L+1. For even n, L = (n^24)/4 and H = L+2.  Richard R. Forberg, May 31 2013


REFERENCES

A. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 116117, 1966.


LINKS

Table of n, a(n) for n=1..99.
H. Becker, Pythagorean triples triplets in JavaScript
Ron Knott, Pythagorean Triples and Online Calculators
Project Euler, Problem 176: Rectangular triangles that share a cathetus.
F. Richman, Pythagorean Triples
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331340. 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


MATHEMATICA

f[n_] := (DivisorSigma[0, If[OddQ[n], n, n / 2]^2]  1) / 2; Table[f[i], {i, 100}] (* Amber Hu (hupo001(AT)gmail.com), Jan 23 2008 *)


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


CROSSREFS

Cf. A046080, A046081, A001227, A018892, A024361, A024362, A024363.
Sequence in context: A078692 A273432 A033151 * A279104 A165509 A100996
Adjacent sequences: A046076 A046077 A046078 * A046080 A046081 A046082


KEYWORD

nonn


AUTHOR

Eric W. Weisstein


STATUS

approved



