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A046079
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Number of Pythagorean triangles with leg n.
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20
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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
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1,8
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COMMENTS
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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(Added by Ant King, 29 Jan 2011).
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REFERENCES
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A. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 116-117, 1966.
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LINKS
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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
Eric Weisstein's World of Mathematics, Pythagorean Triple
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FORMULA
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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
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MATHEMATICA
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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
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PROG
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(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
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CROSSREFS
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Cf. A046080, A046081, A001227, A018892, A024361, A024362, A024363.
Sequence in context: A157654 A078692 A033151 * A165509 A100996 A090048
Adjacent sequences: A046076 A046077 A046078 * A046080 A046081 A046082
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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