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A078926
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Number of primitive Pythagorean triangles with perimeter 2n.
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5
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0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.
Equivalently, number of odd unitary divisors d of n such that sqrt(n) < d < sqrt(2n). (A divisor d of n is 'unitary' if gcd(d,n/d) = 1.) Sketch of proof: A primitive Pythagorean triangle has edge lengths (r^2-s^2, 2rs, r^2+s^2), where 1<=s<r, r and s are relatively prime and r+s is odd. This has perimeter 2n iff n=r(r+s). Let d=r+s.
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EXAMPLE
| a(858)=2; the primitive Pythagorean triangles with edge lengths (364, 627, 725) and (195, 748, 773) both have perimeter 2*858 = 1716.
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MATHEMATICA
| oddpart[n_] := If[OddQ[n], n, oddpart[n/2]]; a[n_] := Length[Select[Divisors[oddpart[n]], n<#^2<2n&&GCD[ #, n/# ]==1&]]
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PROG
| (MAGMA) UnitaryDivisors :=
func<n| [d:d in Divisors(n)|GCD(d, n div d) eq 1]>;
A078926:=
func<n | #[d:d in UnitaryDivisors(n)| IsOdd(d) and Isqrt(n) lt d and d le Isqrt(2*n-1)] >;
[A078926(n):n in [1..105]];
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CROSSREFS
| a(n) = A070109(2n). A078927(n) is smallest s such that a(s)=n. a(n) is nonzero iff n is in A020886.
Sequence in context: A011678 A011679 A011682 * A025458 A179527 A172051
Adjacent sequences: A078923 A078924 A078925 * A078927 A078928 A078929
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KEYWORD
| nonn,easy
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AUTHOR
| Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 15 2002
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