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A078926
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Number of primitive Pythagorean triangles with perimeter 2n.
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6
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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
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OFFSET
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1,858
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COMMENTS
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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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LINKS
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EXAMPLE
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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
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oddpart[n_] := If[OddQ[n], n, oddpart[n/2]]; a[n_] := Length[Select[Divisors[oddpart[n]], n<#^2<2n&&GCD[ #, n/# ]==1&]]
(* Second program: *)
Table[DivisorSum[n/2^IntegerExponent[n, 2], 1 &, n < #^2 < 2 n && CoprimeQ[#, n/#] &], {n, 105}] (* Michael De Vlieger, Oct 08 2017 *)
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PROG
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(Magma) UnitaryDivisors :=
func<n| [d:d in Divisors(n)|GCD(d, n div d) eq 1]>;
func<n | #[d:d in UnitaryDivisors(n)| IsOdd(d) and Isqrt(n) lt d and d le Isqrt(2*n-1)] >;
(PARI) A078926(n) = sumdiv(n, d, (d%2)*(1==gcd(d, n/d))*((d*d)>n)*((d*d)<(2*n))); \\ Antti Karttunen, Oct 07 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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