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Search: id:A156684
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| A156684 |
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The number of primitive Pythagorean triples A^2+B^2=C^2 with 0<A<B<C and gcd(A,B)=1, and both legs less than n. |
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+0
1
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| 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8
(list; graph; listen)
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OFFSET
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1,13
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COMMENT
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For large N, Benito and Varona have shown that a(N)~2/pi^2 Log(1+sqrt(2)).N +O(sqrt(N)). However, the approximations to a(N)/N are considerably more accurate than the error term suggests, and it certainly appears that the density of the primitive triples with both legs less than N tends towards 2/pi^2 Log(1+sqrt(2))=0.1786... as N becomes large.
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REFERENCES
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Benito, Manuel and Varona, Juan; Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics 143, (2002), pp. 117-126.
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EXAMPLE
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There are two primitive triples with both legs less than 14, specifically (3,4,5) and (5,12,13). Hence a(14)=2.
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MATHEMATICA
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PrimitivePythagoreanTriplets[n_]:=Module[{t={{3, 4, 5}}, i=4, j=5}, While[i<n, If[GCD[i, j]==1, h=Sqrt[i^2+j^2]; If[IntegerQ[h] && j<n, AppendTo[t, {i, j, h}]]; ]; If[j<n, j+=2, i++; j=i+1]]; t]; Append[{0, 0, 0, 0}, Length[PrimitivePythagoreanTriplets[ # ]]&/@Range[5, 50]]//Flatten
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CROSSREFS
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A024361
Sequence in context: A106432 A029836 A004257 this_sequence A070564 A072358 A074795
Adjacent sequences: A156681 A156682 A156683 this_sequence A156685 A156686 A156687
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Ant King (mathstutoring(AT)ntlworld.com), Feb 17 2009
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