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A334177
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Squarefree part of numerator of the squared area of the Heronian triangle with sequential prime sides whose shortest leg is prime(n).
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2
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0, 3, 299, 2635, 81795, 3795, 16107, 30459, 64491, 3045315, 537915, 64155, 24235, 301587, 7995, 32512755, 44450835, 56771715, 72867795, 1116115, 111121395, 144843315, 7564011, 247292115, 33656315, 345261315, 382278435, 432567555, 534226515, 76227515, 7407435, 1012520355, 978027, 1365306243
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OFFSET
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1,2
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COMMENTS
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These are the values within the irreducible square roots, which are part of the area resulting from Heronian triangles, which have all sides being sequential prime numbers. The triangles follow this sequence of sides: {prime(n), prime(n+1), prime(n+2)} and their area is represented by {(r/s)*sqrt(t)}, where r, s, t are integers and a(n) is the number t.
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LINKS
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EXAMPLE
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a(1) = 0, since the triangle {2,3,5} has area 0. a(2) = 3, because the first possible Heronian triangle with sequential prime sides and positive area is the triangle {3,5,7} and its respective area is {15*sqrt(3)/4}. a(3) = 299 since the second possible triangle is {5,7,11} which has area {3*sqrt(299)/4}. And so on.
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MATHEMATICA
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a[n_]:=Module[{y, z}, If[n==1, 0, z=Area@SSSTriangle[Prime[n], Prime[n+1], Prime[n+2]]; ({z}/.Coefficient[{z}/.Sqrt[_]->y, y][[1]]->1)[[1]]^2]]
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PROG
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(PARI) a(n) = my(p=prime(n), q=nextprime(p+1), r=nextprime(q+1), s=(p+q+r)/2); core(numerator(s*(s-p)*(s-q)*(s-r))); \\ Michel Marcus, Apr 18 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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